[<< wikibooks] Differentiable Manifolds/Bases of tangent and cotangent spaces and the differentials
In this section we shall

give one base for the tangent and cotangent space for each chart at a point of a manifold,
show how to convert representations in one base into another,
define the differentials of functions from a manifold to the real line, from an interval to a manifold and from a manifold to another manifold,
and prove the chain, product and quotient rules for those differentials.


== Some bases of the tangent space ==

In the following, we will show that these functionals are a basis of the tangent space.
Theorem 2.2: Let 
  
    
      
        M
      
    
    {\displaystyle M}
   be a 
  
    
      
        d
      
    
    {\displaystyle d}
  -dimensional manifold of class 
  
    
      
        
          
            
              C
            
          
          
            n
          
        
      
    
    {\displaystyle {\mathcal {C}}^{n}}
   with 
  
    
      
        n
        ≥
        1
      
    
    {\displaystyle n\geq 1}
   and atlas 
  
    
      
        {
        (
        
          O
          
            υ
          
        
        ,
        
          ϕ
          
            υ
          
        
        )
        
          |
        
        υ
        ∈
        Υ
        }
      
    
    {\displaystyle \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}
  , let 
  
    
      
        (
        O
        ,
        ϕ
        )
        ∈
        {
        (
        
          O
          
            υ
          
        
        ,
        
          ϕ
          
            υ
          
        
        )
        
          |
        
        υ
        ∈
        Υ
        }
      
    
    {\displaystyle (O,\phi )\in \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}
   and let 
  
    
      
        p
        ∈
        O
      
    
    {\displaystyle p\in O}
  . For all 
  
    
      
        j
        ∈
        {
        1
        ,
        …
        ,
        d
        }
      
    
    {\displaystyle j\in \{1,\ldots ,d\}}
  :

  
    
      
        
          
            (
            
              
                ∂
                
                  ∂
                  
                    ϕ
                    
                      j
                    
                  
                
              
            
            )
          
          
            p
          
        
        ∈
        
          T
          
            p
          
        
        M
      
    
    {\displaystyle \left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}\in T_{p}M}
  i. e. the function 
  
    
      
        
          
            (
            
              
                ∂
                
                  ∂
                  
                    ϕ
                    
                      j
                    
                  
                
              
            
            )
          
          
            p
          
        
        :
        
          
            
              C
            
          
          
            n
          
        
        (
        M
        )
        →
        
          R
        
      
    
    {\displaystyle \left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}:{\mathcal {C}}^{n}(M)\to \mathbb {R} }
   is contained in the tangent space 
  
    
      
        
          T
          
            p
          
        
        M
      
    
    {\displaystyle T_{p}M}
  .
Proof:
Let 
  
    
      
        φ
        ,
        ϑ
        ∈
        
          
            
              C
            
          
          
            n
          
        
        (
        M
        )
      
    
    {\displaystyle \varphi ,\vartheta \in {\mathcal {C}}^{n}(M)}
  .
1. We show linearity.

  
    
      
        
          
            
              
                
                  
                    (
                    
                      
                        ∂
                        
                          ∂
                          
                            ϕ
                            
                              j
                            
                          
                        
                      
                    
                    )
                  
                  
                    p
                  
                
                (
                φ
                +
                c
                ϑ
                )
              
              
                
                =
                
                  (
                  
                    
                      ∂
                      
                        
                          x
                          
                            j
                          
                        
                      
                    
                    (
                    (
                    φ
                    +
                    c
                    ϑ
                    )
                    ∘
                    
                      ϕ
                      
                        −
                        1
                      
                    
                    )
                  
                  )
                
                (
                ϕ
                (
                p
                )
                )
              
            
            
              
              
                
                =
                
                  (
                  
                    
                      ∂
                      
                        
                          x
                          
                            j
                          
                        
                      
                    
                    (
                    φ
                    ∘
                    
                      ϕ
                      
                        −
                        1
                      
                    
                    +
                    c
                    ϑ
                    ∘
                    
                      ϕ
                      
                        −
                        1
                      
                    
                    )
                  
                  )
                
                (
                ϕ
                (
                p
                )
                )
              
            
            
              
              
                
                =
                
                  (
                  
                    
                      ∂
                      
                        
                          x
                          
                            j
                          
                        
                      
                    
                    (
                    φ
                    ∘
                    
                      ϕ
                      
                        −
                        1
                      
                    
                    )
                    +
                    c
                    
                      ∂
                      
                        
                          x
                          
                            j
                          
                        
                      
                    
                    (
                    ϑ
                    ∘
                    
                      ϕ
                      
                        −
                        1
                      
                    
                    )
                    )
                  
                  )
                
                (
                ϕ
                (
                p
                )
                )
              
            
            
              
              
                
                =
                
                  
                    (
                    
                      
                        ∂
                        
                          ∂
                          
                            ϕ
                            
                              j
                            
                          
                        
                      
                    
                    )
                  
                  
                    p
                  
                
                (
                φ
                )
                +
                c
                
                  
                    (
                    
                      
                        ∂
                        
                          ∂
                          
                            ϕ
                            
                              j
                            
                          
                        
                      
                    
                    )
                  
                  
                    p
                  
                
                (
                ϑ
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\varphi +c\vartheta )&=\left(\partial _{x_{j}}((\varphi +c\vartheta )\circ \phi ^{-1})\right)(\phi (p))\\&=\left(\partial _{x_{j}}(\varphi \circ \phi ^{-1}+c\vartheta \circ \phi ^{-1})\right)(\phi (p))\\&=\left(\partial _{x_{j}}(\varphi \circ \phi ^{-1})+c\partial _{x_{j}}(\vartheta \circ \phi ^{-1}))\right)(\phi (p))\\&=\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\varphi )+c\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\vartheta )\end{aligned}}}
  From the second to the third line, we used the linearity of the derivative.
2. We show the product rule.

  
    
      
        
          
            
              
                
                  
                    (
                    
                      
                        ∂
                        
                          ∂
                          
                            ϕ
                            
                              j
                            
                          
                        
                      
                    
                    )
                  
                  
                    p
                  
                
                (
                φ
                ϑ
                )
              
              
                
                =
                
                  (
                  
                    
                      ∂
                      
                        
                          x
                          
                            j
                          
                        
                      
                    
                    (
                    (
                    φ
                    ϑ
                    )
                    ∘
                    
                      ϕ
                      
                        −
                        1
                      
                    
                    )
                  
                  )
                
                (
                ϕ
                (
                p
                )
                )
              
            
            
              
              
                
                =
                
                  (
                  
                    
                      ∂
                      
                        
                          x
                          
                            j
                          
                        
                      
                    
                    (
                    (
                    φ
                    ∘
                    
                      ϕ
                      
                        −
                        1
                      
                    
                    )
                    (
                    ϑ
                    ∘
                    
                      ϕ
                      
                        −
                        1
                      
                    
                    )
                    )
                  
                  )
                
                (
                ϕ
                (
                p
                )
                )
              
            
            
              
              
                
                =
                (
                φ
                ∘
                
                  ϕ
                  
                    −
                    1
                  
                
                )
                (
                ϕ
                (
                p
                )
                )
                
                  (
                  
                    
                      ∂
                      
                        
                          x
                          
                            j
                          
                        
                      
                    
                    (
                    ϑ
                    ∘
                    
                      ϕ
                      
                        −
                        1
                      
                    
                    )
                  
                  )
                
                (
                ϕ
                (
                p
                )
                )
                +
                (
                ϑ
                ∘
                
                  ϕ
                  
                    −
                    1
                  
                
                )
                (
                ϕ
                (
                p
                )
                )
                
                  (
                  
                    
                      ∂
                      
                        
                          x
                          
                            j
                          
                        
                      
                    
                    (
                    φ
                    ∘
                    
                      ϕ
                      
                        −
                        1
                      
                    
                    )
                  
                  )
                
                (
                ϕ
                (
                p
                )
                )
              
            
            
              
              
                
                =
                φ
                (
                p
                )
                
                  
                    (
                    
                      
                        ∂
                        
                          ∂
                          
                            ϕ
                            
                              j
                            
                          
                        
                      
                    
                    )
                  
                  
                    p
                  
                
                (
                ϑ
                )
                +
                ϑ
                (
                p
                )
                
                  
                    (
                    
                      
                        ∂
                        
                          ∂
                          
                            ϕ
                            
                              j
                            
                          
                        
                      
                    
                    )
                  
                  
                    p
                  
                
                (
                φ
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\varphi \vartheta )&=\left(\partial _{x_{j}}((\varphi \vartheta )\circ \phi ^{-1})\right)(\phi (p))\\&=\left(\partial _{x_{j}}((\varphi \circ \phi ^{-1})(\vartheta \circ \phi ^{-1}))\right)(\phi (p))\\&=(\varphi \circ \phi ^{-1})(\phi (p))\left(\partial _{x_{j}}(\vartheta \circ \phi ^{-1})\right)(\phi (p))+(\vartheta \circ \phi ^{-1})(\phi (p))\left(\partial _{x_{j}}(\varphi \circ \phi ^{-1})\right)(\phi (p))\\&=\varphi (p)\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\vartheta )+\vartheta (p)\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\varphi )\end{aligned}}}
  From the second to the third line, we used the product rule of the derivative.
3. It follows from the definition of 
  
    
      
        
          
            (
            
              
                ∂
                
                  ∂
                  
                    ϕ
                    
                      j
                    
                  
                
              
            
            )
          
          
            p
          
        
      
    
    {\displaystyle \left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}}
  , that 
  
    
      
        
          
            (
            
              
                ∂
                
                  ∂
                  
                    ϕ
                    
                      j
                    
                  
                
              
            
            )
          
          
            p
          
        
        (
        φ
        )
        =
        0
      
    
    {\displaystyle \left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\varphi )=0}
   if 
  
    
      
        φ
      
    
    {\displaystyle \varphi }
   is not defined at 
  
    
      
        p
      
    
    {\displaystyle p}
  .
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  
Lemma 2.3: Let 
  
    
      
        M
      
    
    {\displaystyle M}
   be a 
  
    
      
        d
      
    
    {\displaystyle d}
  -dimensional manifold of class 
  
    
      
        
          
            
              C
            
          
          
            n
          
        
      
    
    {\displaystyle {\mathcal {C}}^{n}}
   with atlas 
  
    
      
        {
        (
        
          O
          
            υ
          
        
        ,
        
          ϕ
          
            υ
          
        
        )
        
          |
        
        υ
        ∈
        Υ
        }
      
    
    {\displaystyle \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}
  , and let 
  
    
      
        (
        O
        ,
        ϕ
        )
        ∈
        {
        (
        
          O
          
            υ
          
        
        ,
        
          ϕ
          
            υ
          
        
        )
        
          |
        
        υ
        ∈
        Υ
        }
      
    
    {\displaystyle (O,\phi )\in \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}
  . If we write 
  
    
      
        ϕ
        =
        (
        
          ϕ
          
            1
          
        
        ,
        …
        ,
        
          ϕ
          
            d
          
        
        )
      
    
    {\displaystyle \phi =(\phi _{1},\ldots ,\phi _{d})}
  , then we have for each 
  
    
      
        k
        ∈
        {
        1
        ,
        …
        ,
        d
        }
      
    
    {\displaystyle k\in \{1,\ldots ,d\}}
  , that 
  
    
      
        
          ϕ
          
            k
          
        
        ∈
        
          
            
              C
            
          
          
            n
          
        
        (
        M
        )
      
    
    {\displaystyle \phi _{k}\in {\mathcal {C}}^{n}(M)}
  .
Proof:
Let 
  
    
      
        (
        U
        ,
        θ
        )
        ∈
        {
        (
        
          O
          
            υ
          
        
        ,
        
          ϕ
          
            υ
          
        
        )
        
          |
        
        υ
        ∈
        Υ
        }
      
    
    {\displaystyle (U,\theta )\in \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}
  . Since 
  
    
      
        {
        (
        
          O
          
            υ
          
        
        ,
        
          ϕ
          
            υ
          
        
        )
        
          |
        
        υ
        ∈
        Υ
        }
      
    
    {\displaystyle \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}
   is an atlas, 
  
    
      
        θ
      
    
    {\displaystyle \theta }
   and 
  
    
      
        ϕ
      
    
    {\displaystyle \phi }
   are compatible. From this follows that the function

  
    
      
        ϕ
        
          
            |
          
          
            U
            ∩
            O
          
        
        ∘
        θ
        
          
            |
          
          
            O
            ∩
            U
          
          
            −
            1
          
        
      
    
    {\displaystyle \phi |_{U\cap O}\circ \theta |_{O\cap U}^{-1}}
  is of class 
  
    
      
        
          
            
              C
            
          
          
            n
          
        
      
    
    {\displaystyle {\mathcal {C}}^{n}}
  . But if we denote by 
  
    
      
        
          π
          
            k
          
        
      
    
    {\displaystyle \pi _{k}}
   the function

  
    
      
        
          π
          
            k
          
        
        :
        
          
            R
          
          
            d
          
        
        →
        
          R
        
        ,
        
          π
          
            k
          
        
        (
        
          x
          
            1
          
        
        ,
        …
        ,
        
          x
          
            d
          
        
        )
        =
        
          x
          
            k
          
        
      
    
    {\displaystyle \pi _{k}:\mathbb {R} ^{d}\to \mathbb {R} ,\pi _{k}(x_{1},\ldots ,x_{d})=x_{k}}
  , which is also called the projection to the 
  
    
      
        k
      
    
    {\displaystyle k}
  -th component, then we have:

  
    
      
        
          ϕ
          
            k
          
        
        
          
            |
          
          
            U
            ∩
            O
          
        
        ∘
        θ
        
          
            |
          
          
            O
            ∩
            U
          
          
            −
            1
          
        
        =
        
          π
          
            k
          
        
        ∘
        ϕ
        
          
            |
          
          
            U
            ∩
            O
          
        
        ∘
        θ
        
          
            |
          
          
            O
            ∩
            U
          
          
            −
            1
          
        
      
    
    {\displaystyle \phi _{k}|_{U\cap O}\circ \theta |_{O\cap U}^{-1}=\pi _{k}\circ \phi |_{U\cap O}\circ \theta |_{O\cap U}^{-1}}
  It is not difficult to show that 
  
    
      
        
          π
          
            k
          
        
      
    
    {\displaystyle \pi _{k}}
   is contained in 
  
    
      
        
          
            
              C
            
          
          
            ∞
          
        
        (
        
          
            R
          
          
            d
          
        
        ,
        
          R
        
        )
      
    
    {\displaystyle {\mathcal {C}}^{\infty }(\mathbb {R} ^{d},\mathbb {R} )}
  , and therefore the function

  
    
      
        
          π
          
            k
          
        
        ∘
        ϕ
        
          
            |
          
          
            U
            ∩
            O
          
        
        ∘
        θ
        
          
            |
          
          
            O
            ∩
            U
          
          
            −
            1
          
        
      
    
    {\displaystyle \pi _{k}\circ \phi |_{U\cap O}\circ \theta |_{O\cap U}^{-1}}
  is contained in 
  
    
      
        
          
            
              C
            
          
          
            n
          
        
        (
        
          
            R
          
          
            d
          
        
        ,
        
          R
        
        )
      
    
    {\displaystyle {\mathcal {C}}^{n}(\mathbb {R} ^{d},\mathbb {R} )}
   as a composition of 
  
    
      
        n
      
    
    {\displaystyle n}
  -times continuously differentiable functions (or continuous functions if 
  
    
      
        n
        =
        0
      
    
    {\displaystyle n=0}
  ).
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  
Lemma 2.4: Let 
  
    
      
        M
      
    
    {\displaystyle M}
   be a 
  
    
      
        d
      
    
    {\displaystyle d}
  -dimensional manifold of class 
  
    
      
        
          
            
              C
            
          
          
            n
          
        
      
    
    {\displaystyle {\mathcal {C}}^{n}}
   with 
  
    
      
        n
        ≥
        1
      
    
    {\displaystyle n\geq 1}
   and atlas 
  
    
      
        {
        (
        
          O
          
            υ
          
        
        ,
        
          ϕ
          
            υ
          
        
        )
        
          |
        
        υ
        ∈
        Υ
        }
      
    
    {\displaystyle \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}
  , let 
  
    
      
        (
        O
        ,
        ϕ
        )
        ∈
        {
        (
        
          O
          
            υ
          
        
        ,
        
          ϕ
          
            υ
          
        
        )
        
          |
        
        υ
        ∈
        Υ
        }
      
    
    {\displaystyle (O,\phi )\in \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}
   and let 
  
    
      
        p
        ∈
        O
      
    
    {\displaystyle p\in O}
  . If we write 
  
    
      
        ϕ
        =
        (
        
          ϕ
          
            1
          
        
        ,
        …
        ,
        
          ϕ
          
            d
          
        
        )
      
    
    {\displaystyle \phi =(\phi _{1},\ldots ,\phi _{d})}
   we have:

  
    
      
        
          
            (
            
              
                ∂
                
                  ∂
                  
                    ϕ
                    
                      j
                    
                  
                
              
            
            )
          
          
            p
          
        
        (
        
          ϕ
          
            k
          
        
        )
        =
        
          
            {
            
              
                
                  1
                
                
                  j
                  =
                  k
                
              
              
                
                  0
                
                
                  j
                  ≠
                  k
                
              
            
            
          
        
      
    
    {\displaystyle \left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\phi _{k})={\begin{cases}1&j=k\\0&j\neq k\end{cases}}}
  Note that due to lemma 2.3, 
  
    
      
        
          ϕ
          
            k
          
        
        ∈
        
          
            
              C
            
          
          
            n
          
        
        (
        M
        )
      
    
    {\displaystyle \phi _{k}\in {\mathcal {C}}^{n}(M)}
   for all 
  
    
      
        k
        ∈
        {
        1
        ,
        …
        ,
        d
        }
      
    
    {\displaystyle k\in \{1,\ldots ,d\}}
  , which is why the above expression makes sense.
Proof:
We have:

  
    
      
        
          
            
              
                
                  
                    (
                    
                      
                        ∂
                        
                          ∂
                          
                            ϕ
                            
                              j
                            
                          
                        
                      
                    
                    )
                  
                  
                    p
                  
                
                (
                
                  ϕ
                  
                    k
                  
                
                )
              
              
                
                =
                
                  (
                  
                    
                      ∂
                      
                        
                          x
                          
                            j
                          
                        
                      
                    
                    (
                    
                      ϕ
                      
                        k
                      
                    
                    ∘
                    
                      ϕ
                      
                        −
                        1
                      
                    
                    )
                  
                  )
                
                (
                ϕ
                (
                p
                )
                )
              
            
            
              
              
                
                =
                
                  lim
                  
                    y
                    →
                    0
                  
                
                
                  
                    
                      (
                      
                        ϕ
                        
                          k
                        
                      
                      ∘
                      
                        ϕ
                        
                          −
                          1
                        
                      
                      )
                      (
                      
                        x
                        
                          1
                        
                      
                      ,
                      …
                      ,
                      
                        x
                        
                          j
                          −
                          1
                        
                      
                      ,
                      
                        x
                        
                          j
                        
                      
                      +
                      y
                      ,
                      
                        x
                        
                          j
                          +
                          1
                        
                      
                      ,
                      …
                      ,
                      
                        x
                        
                          d
                        
                      
                      )
                      −
                      (
                      
                        ϕ
                        
                          k
                        
                      
                      ∘
                      
                        ϕ
                        
                          −
                          1
                        
                      
                      )
                      (
                      
                        x
                        
                          1
                        
                      
                      ,
                      …
                      ,
                      
                        x
                        
                          d
                        
                      
                      )
                    
                    y
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\phi _{k})&=\left(\partial _{x_{j}}(\phi _{k}\circ \phi ^{-1})\right)(\phi (p))\\&=\lim _{y\to 0}{\frac {(\phi _{k}\circ \phi ^{-1})(x_{1},\ldots ,x_{j-1},x_{j}+y,x_{j+1},\ldots ,x_{d})-(\phi _{k}\circ \phi ^{-1})(x_{1},\ldots ,x_{d})}{y}}\end{aligned}}}
  Further,

  
    
      
        (
        
          ϕ
          
            k
          
        
        ∘
        
          ϕ
          
            −
            1
          
        
        )
        (
        
          x
          
            1
          
        
        ,
        …
        ,
        
          x
          
            d
          
        
        )
        =
        
          x
          
            k
          
        
      
    
    {\displaystyle (\phi _{k}\circ \phi ^{-1})(x_{1},\ldots ,x_{d})=x_{k}}
  and

  
    
      
        (
        
          ϕ
          
            k
          
        
        ∘
        
          ϕ
          
            −
            1
          
        
        )
        (
        
          x
          
            1
          
        
        ,
        …
        ,
        
          x
          
            j
            −
            1
          
        
        ,
        
          x
          
            j
          
        
        +
        y
        ,
        
          x
          
            j
            +
            1
          
        
        ,
        …
        ,
        
          x
          
            d
          
        
        )
        =
        
          
            {
            
              
                
                  
                    x
                    
                      k
                    
                  
                  +
                  y
                
                
                  k
                  =
                  j
                
              
              
                
                  
                    x
                    
                      k
                    
                  
                
                
                  k
                  ≠
                  j
                
              
            
            
          
        
      
    
    {\displaystyle (\phi _{k}\circ \phi ^{-1})(x_{1},\ldots ,x_{j-1},x_{j}+y,x_{j+1},\ldots ,x_{d})={\begin{cases}x_{k}+y&k=j\\x_{k}&k\neq j\end{cases}}}
  Inserting this in the above limit gives the lemma.
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  
Theorem 2.5: Let 
  
    
      
        M
      
    
    {\displaystyle M}
   be a 
  
    
      
        d
      
    
    {\displaystyle d}
  -dimensional manifold of class 
  
    
      
        
          
            
              C
            
          
          
            n
          
        
      
    
    {\displaystyle {\mathcal {C}}^{n}}
   with 
  
    
      
        n
        ≥
        1
      
    
    {\displaystyle n\geq 1}
   and atlas 
  
    
      
        {
        (
        
          O
          
            υ
          
        
        ,
        
          ϕ
          
            υ
          
        
        )
        
          |
        
        υ
        ∈
        Υ
        }
      
    
    {\displaystyle \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}
  , let 
  
    
      
        (
        O
        ,
        ϕ
        )
        ∈
        {
        (
        
          O
          
            υ
          
        
        ,
        
          ϕ
          
            υ
          
        
        )
        
          |
        
        υ
        ∈
        Υ
        }
      
    
    {\displaystyle (O,\phi )\in \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}
   and let 
  
    
      
        p
        ∈
        O
      
    
    {\displaystyle p\in O}
  . The tangent vectors

  
    
      
        
          
            (
            
              
                ∂
                
                  ∂
                  
                    ϕ
                    
                      j
                    
                  
                
              
            
            )
          
          
            p
          
        
        ∈
        
          T
          
            p
          
        
        M
        ,
        j
        ∈
        {
        1
        ,
        …
        ,
        d
        }
      
    
    {\displaystyle \left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}\in T_{p}M,j\in \{1,\ldots ,d\}}
  are linearly independent.
Proof:
We write again 
  
    
      
        ϕ
        =
        (
        
          ϕ
          
            1
          
        
        ,
        …
        ,
        
          ϕ
          
            d
          
        
        )
      
    
    {\displaystyle \phi =(\phi _{1},\ldots ,\phi _{d})}
  .
Let 
  
    
      
        
          ∑
          
            j
            =
            1
          
          
            d
          
        
        
          a
          
            j
          
        
        
          
            (
            
              
                ∂
                
                  ∂
                  
                    ϕ
                    
                      j
                    
                  
                
              
            
            )
          
          
            p
          
        
        =
        
          0
          
            p
          
        
      
    
    {\displaystyle \sum _{j=1}^{d}a_{j}\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}=0_{p}}
  . Then we have for all 
  
    
      
        k
        ∈
        {
        1
        ,
        …
        ,
        d
        }
      
    
    {\displaystyle k\in \{1,\ldots ,d\}}
  :

  
    
      
        0
        =
        
          0
          
            p
          
        
        (
        
          ϕ
          
            k
          
        
        )
        =
        
          ∑
          
            j
            =
            1
          
          
            d
          
        
        
          a
          
            j
          
        
        
          
            (
            
              
                ∂
                
                  ∂
                  
                    ϕ
                    
                      j
                    
                  
                
              
            
            )
          
          
            p
          
        
        (
        
          ϕ
          
            k
          
        
        )
        =
        
          a
          
            k
          
        
      
    
    {\displaystyle 0=0_{p}(\phi _{k})=\sum _{j=1}^{d}a_{j}\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\phi _{k})=a_{k}}
  
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  Lemma 2.6:
Let 
  
    
      
        M
      
    
    {\displaystyle M}
   be a manifold with atlas 
  
    
      
        {
        (
        
          O
          
            υ
          
        
        ,
        
          ϕ
          
            υ
          
        
        )
        
          |
        
        υ
        ∈
        Υ
        }
      
    
    {\displaystyle \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}
  , 
  
    
      
        p
        ∈
        M
      
    
    {\displaystyle p\in M}
  , 
  
    
      
        V
        ⊆
        M
      
    
    {\displaystyle V\subseteq M}
   be open, let 
  
    
      
        
          
            V
          
          
            p
          
        
        ∈
        
          T
          
            p
          
        
        M
      
    
    {\displaystyle \mathbf {V} _{p}\in T_{p}M}
   and 
  
    
      
        φ
        :
        V
        →
        
          R
        
        ,
        φ
        (
        q
        )
        =
        c
      
    
    {\displaystyle \varphi :V\to \mathbb {R} ,\varphi (q)=c}
   for a 
  
    
      
        c
        ∈
        
          R
        
      
    
    {\displaystyle c\in \mathbb {R} }
  ; i. e. 
  
    
      
        φ
      
    
    {\displaystyle \varphi }
   is a constant function. Then 
  
    
      
        φ
        ∈
        
          
            
              C
            
          
          
            ∞
          
        
        (
        M
        )
      
    
    {\displaystyle \varphi \in {\mathcal {C}}^{\infty }(M)}
   and 
  
    
      
        
          
            V
          
          
            p
          
        
        (
        φ
        )
        =
        0
      
    
    {\displaystyle \mathbf {V} _{p}(\varphi )=0}
  .
Proof:
1. We show 
  
    
      
        φ
        ∈
        
          
            
              C
            
          
          
            ∞
          
        
        (
        M
        )
      
    
    {\displaystyle \varphi \in {\mathcal {C}}^{\infty }(M)}
  .
By assumption, 
  
    
      
        V
        ⊆
        M
      
    
    {\displaystyle V\subseteq M}
   is open. This means the first part of the definition of a 
  
    
      
        
          
            
              C
            
          
          
            ∞
          
        
        (
        M
        )
      
    
    {\displaystyle {\mathcal {C}}^{\infty }(M)}
   is fulfilled.
Further, for each 
  
    
      
        (
        U
        ,
        θ
        )
        ∈
        {
        (
        
          O
          
            υ
          
        
        ,
        
          ϕ
          
            υ
          
        
        )
        
          |
        
        υ
        ∈
        Υ
        }
      
    
    {\displaystyle (U,\theta )\in \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}
   and 
  
    
      
        x
        ∈
        θ
        (
        V
        ∩
        U
        )
      
    
    {\displaystyle x\in \theta (V\cap U)}
  , we have:

  
    
      
        φ
        ∘
        θ
        
          
            |
          
          
            U
            ∩
            V
          
        
        (
        x
        )
        =
        c
      
    
    {\displaystyle \varphi \circ \theta |_{U\cap V}(x)=c}
  This is contained in 
  
    
      
        
          
            
              C
            
          
          
            ∞
          
        
        (
        
          
            R
          
          
            d
          
        
        ,
        
          R
        
        )
      
    
    {\displaystyle {\mathcal {C}}^{\infty }(\mathbb {R} ^{d},\mathbb {R} )}
  .
2. We show that 
  
    
      
        
          
            V
          
          
            p
          
        
        (
        φ
        )
        =
        0
      
    
    {\displaystyle \mathbf {V} _{p}(\varphi )=0}
  .
We define 
  
    
      
        ϑ
        :
        V
        →
        
          R
        
        ,
        ϑ
        (
        q
        )
        =
        1
      
    
    {\displaystyle \vartheta :V\to \mathbb {R} ,\vartheta (q)=1}
  . Using the two rules linearity and product rule for tangent vectors, we obtain:

  
    
      
        
          
            V
          
          
            p
          
        
        (
        φ
        )
        =
        
          
            V
          
          
            p
          
        
        (
        ϑ
        φ
        )
        =
        1
        
          
            V
          
          
            p
          
        
        (
        φ
        )
        +
        φ
        (
        p
        )
        
          
            V
          
          
            p
          
        
        (
        ϑ
        )
        =
        
          
            V
          
          
            p
          
        
        (
        φ
        )
        +
        
          
            V
          
          
            p
          
        
        (
        ϑ
        φ
        (
        p
        )
        )
        =
        
          
            V
          
          
            p
          
        
        (
        φ
        )
        +
        
          
            V
          
          
            p
          
        
        (
        φ
        )
      
    
    {\displaystyle \mathbf {V} _{p}(\varphi )=\mathbf {V} _{p}(\vartheta \varphi )=1\mathbf {V} _{p}(\varphi )+\varphi (p)\mathbf {V} _{p}(\vartheta )=\mathbf {V} _{p}(\varphi )+\mathbf {V} _{p}(\vartheta \varphi (p))=\mathbf {V} _{p}(\varphi )+\mathbf {V} _{p}(\varphi )}
  Substracting 
  
    
      
        
          
            V
          
          
            p
          
        
        (
        φ
        )
      
    
    {\displaystyle \mathbf {V} _{p}(\varphi )}
  , we obtain 
  
    
      
        
          
            V
          
          
            p
          
        
        (
        φ
        )
        =
        0
      
    
    {\displaystyle \mathbf {V} _{p}(\varphi )=0}
  .
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  

Proof:
Let 
  
    
      
        U
        ⊆
        M
      
    
    {\displaystyle U\subseteq M}
   be open, and let 
  
    
      
        φ
        :
        U
        →
        
          R
        
      
    
    {\displaystyle \varphi :U\to \mathbb {R} }
   be contained in 
  
    
      
        
          
            
              C
            
          
          
            n
          
        
        (
        M
        )
      
    
    {\displaystyle {\mathcal {C}}^{n}(M)}
  .
Case 1: 
  
    
      
        p
        ∉
        U
      
    
    {\displaystyle p\notin U}
  .
In this case, 
  
    
      
        
          
            V
          
          
            p
          
        
        (
        φ
        )
        =
        0
      
    
    {\displaystyle \mathbf {V} _{p}(\varphi )=0}
   and 
  
    
      
        
          
            (
            
              
                ∂
                
                  ∂
                  
                    ϕ
                    
                      j
                    
                  
                
              
            
            )
          
          
            p
          
        
        (
        φ
        )
        =
        0
      
    
    {\displaystyle \left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\varphi )=0}
  , since 
  
    
      
        φ
      
    
    {\displaystyle \varphi }
   is not defined at 
  
    
      
        p
      
    
    {\displaystyle p}
   and both 
  
    
      
        
          
            V
          
          
            p
          
        
      
    
    {\displaystyle \mathbf {V} _{p}}
   and 
  
    
      
        
          
            (
            
              
                ∂
                
                  ∂
                  
                    ϕ
                    
                      j
                    
                  
                
              
            
            )
          
          
            p
          
        
      
    
    {\displaystyle \left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}}
   are tangent vectors. From this follows the formula.
Case 2: 
  
    
      
        p
        ∈
        U
      
    
    {\displaystyle p\in U}
  .
In this case, we obtain that the set 
  
    
      
        ϕ
        (
        U
        ∩
        O
        )
      
    
    {\displaystyle \phi (U\cap O)}
   is open in 
  
    
      
        
          
            R
          
          
            d
          
        
      
    
    {\displaystyle \mathbb {R} ^{d}}
   as follows: Since 
  
    
      
        ϕ
        :
        O
        →
        ϕ
        (
        O
        )
      
    
    {\displaystyle \phi :O\to \phi (O)}
   is a homeomorphism by definition of charts, the set 
  
    
      
        ϕ
        (
        U
        ∩
        O
        )
      
    
    {\displaystyle \phi (U\cap O)}
   is open in 
  
    
      
        ϕ
        (
        O
        )
      
    
    {\displaystyle \phi (O)}
  . By definition of the subspace topology, we have 
  
    
      
        ϕ
        (
        U
        ∩
        O
        )
        =
        V
        ∩
        ϕ
        (
        O
        )
      
    
    {\displaystyle \phi (U\cap O)=V\cap \phi (O)}
   for a 
  
    
      
        V
      
    
    {\displaystyle V}
   open in 
  
    
      
        
          
            R
          
          
            d
          
        
      
    
    {\displaystyle \mathbb {R} ^{d}}
  . But 
  
    
      
        V
        ∩
        ϕ
        (
        O
        )
      
    
    {\displaystyle V\cap \phi (O)}
   is open in 
  
    
      
        
          
            R
          
          
            d
          
        
      
    
    {\displaystyle \mathbb {R} ^{d}}
   as the intersection of two open sets; recall that 
  
    
      
        ϕ
        (
        O
        )
      
    
    {\displaystyle \phi (O)}
   was required to be open in the definition of a chart.
Furthermore, from 
  
    
      
        p
        ∈
        U
      
    
    {\displaystyle p\in U}
   and 
  
    
      
        p
        ∈
        O
      
    
    {\displaystyle p\in O}
   it follows that 
  
    
      
        p
        ∈
        U
        ∩
        O
      
    
    {\displaystyle p\in U\cap O}
  , and therefore 
  
    
      
        ϕ
        (
        p
        )
        ∈
        ϕ
        (
        O
        ∩
        U
        )
      
    
    {\displaystyle \phi (p)\in \phi (O\cap U)}
  . Since 
  
    
      
        ϕ
        (
        O
        ∩
        U
        )
      
    
    {\displaystyle \phi (O\cap U)}
   is open, we find an 
  
    
      
        ϵ
        >
        0
      
    
    {\displaystyle \epsilon >0}
   such that the open ball 
  
    
      
        
          B
          
            ϵ
          
        
        (
        ϕ
        (
        p
        )
        )
      
    
    {\displaystyle B_{\epsilon }(\phi (p))}
   is contained in 
  
    
      
        ϕ
        (
        O
        ∩
        U
        )
      
    
    {\displaystyle \phi (O\cap U)}
  . We define 
  
    
      
        W
        :=
        
          ϕ
          
            −
            1
          
        
        (
        
          B
          
            ϵ
          
        
        (
        ϕ
        (
        p
        )
        )
        )
      
    
    {\displaystyle W:=\phi ^{-1}(B_{\epsilon }(\phi (p)))}
  . Since 
  
    
      
        ϕ
      
    
    {\displaystyle \phi }
   is bijective, 
  
    
      
        W
        ⊆
        U
        ∩
        O
      
    
    {\displaystyle W\subseteq U\cap O}
  , and since 
  
    
      
        ϕ
      
    
    {\displaystyle \phi }
   is a homeomorphism, in particular continuous, 
  
    
      
        W
      
    
    {\displaystyle W}
   is open in 
  
    
      
        O
      
    
    {\displaystyle O}
   with respect to the subspace topology of 
  
    
      
        O
      
    
    {\displaystyle O}
  . From this also follows 
  
    
      
        O
      
    
    {\displaystyle O}
   open in 
  
    
      
        M
      
    
    {\displaystyle M}
  , because if 
  
    
      
        W
      
    
    {\displaystyle W}
   is open in 
  
    
      
        O
      
    
    {\displaystyle O}
  , then by definition of the subspace topology it is of the form 
  
    
      
        V
        ∩
        O
      
    
    {\displaystyle V\cap O}
   for an open set 
  
    
      
        V
        ⊆
        M
      
    
    {\displaystyle V\subseteq M}
  , and hence it is open as the intersection of two open sets.
We have that 
  
    
      
        φ
        
          
            |
          
          
            W
          
        
        :
        W
        →
        
          R
        
      
    
    {\displaystyle \varphi |_{W}:W\to \mathbb {R} }
  , is contained in 
  
    
      
        
          
            
              C
            
          
          
            ∞
          
        
        (
        M
        )
      
    
    {\displaystyle {\mathcal {C}}^{\infty }(M)}
  : 
  
    
      
        W
      
    
    {\displaystyle W}
   is an open subset of 
  
    
      
        M
      
    
    {\displaystyle M}
  , and if 
  
    
      
        (
        V
        ,
        θ
        )
        ∈
        {
        (
        
          O
          
            υ
          
        
        ,
        
          ϕ
          
            υ
          
        
        )
        
          |
        
        υ
        ∈
        Υ
        }
      
    
    {\displaystyle (V,\theta )\in \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}
  , then

  
    
      
        φ
        
          
            |
          
          
            W
            ∩
            V
          
        
        ∘
        θ
        
          
            |
          
          
            W
            ∩
            V
          
          
            −
            1
          
        
        =
        (
        φ
        
          
            |
          
          
            U
            ∩
            V
          
        
        ∘
        θ
        
          
            |
          
          
            U
            ∩
            V
          
          
            −
            1
          
        
        )
        
          
            |
          
          
            θ
            (
            W
            ∩
            V
            )
          
        
      
    
    {\displaystyle \varphi |_{W\cap V}\circ \theta |_{W\cap V}^{-1}=(\varphi |_{U\cap V}\circ \theta |_{U\cap V}^{-1})|_{\theta (W\cap V)}}
  ,(check this by direct calculation!), which is contained in 
  
    
      
        
          
            
              C
            
          
          
            ∞
          
        
        (
        
          
            R
          
          
            d
          
        
        ,
        
          R
        
        )
      
    
    {\displaystyle {\mathcal {C}}^{\infty }(\mathbb {R} ^{d},\mathbb {R} )}
   as the restriction of an arbitrarily often continuously differentiable function.
We now define the function 
  
    
      
        F
        :
        
          B
          
            ϵ
          
        
        (
        ϕ
        (
        p
        )
        )
        →
        
          R
        
      
    
    {\displaystyle F:B_{\epsilon }(\phi (p))\to \mathbb {R} }
  , 
  
    
      
        F
        (
        x
        )
        =
        (
        φ
        ∘
        
          ϕ
          
            −
            1
          
        
        )
        (
        x
        )
      
    
    {\displaystyle F(x)=(\varphi \circ \phi ^{-1})(x)}
  , and further for each 
  
    
      
        x
        ∈
        
          B
          
            ϵ
          
        
        (
        ϕ
        (
        p
        )
        )
      
    
    {\displaystyle x\in B_{\epsilon }(\phi (p))}
  , we define

  
    
      
        
          μ
          
            x
          
        
        (
        ξ
        )
        :=
        F
        (
        ξ
        x
        +
        (
        1
        −
        ξ
        )
        ϕ
        (
        p
        )
        )
      
    
    {\displaystyle \mu _{x}(\xi ):=F(\xi x+(1-\xi )\phi (p))}
  From the fundamental theorem of calculus, the multi-dimensional chain rule and the linearity of the integral follows for each 
  
    
      
        x
        ∈
        
          B
          
            ϵ
          
        
        (
        ϕ
        (
        p
        )
        )
      
    
    {\displaystyle x\in B_{\epsilon }(\phi (p))}
  , that

  
    
      
        
          
            
              
                F
                (
                x
                )
              
              
                
                =
                
                  μ
                  
                    x
                  
                
                (
                1
                )
              
            
            
              
              
                
                =
                
                  μ
                  
                    x
                  
                
                (
                0
                )
                +
                
                  ∫
                  
                    0
                  
                  
                    1
                  
                
                
                  μ
                  
                    x
                  
                  ′
                
                (
                ξ
                )
                d
                ξ
              
            
            
              
              
                
                =
                F
                (
                ϕ
                (
                p
                )
                )
                +
                
                  ∑
                  
                    j
                    =
                    1
                  
                  
                    d
                  
                
                (
                
                  x
                  
                    j
                  
                
                −
                ϕ
                (
                p
                
                  )
                  
                    j
                  
                
                )
                
                  ∫
                  
                    0
                  
                  
                    1
                  
                
                
                  ∂
                  
                    
                      x
                      
                        j
                      
                    
                  
                
                F
                (
                ξ
                ϕ
                (
                p
                )
                +
                (
                1
                −
                ξ
                )
                x
                )
                d
                ξ
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}F(x)&=\mu _{x}(1)\\&=\mu _{x}(0)+\int _{0}^{1}\mu _{x}'(\xi )d\xi \\&=F(\phi (p))+\sum _{j=1}^{d}(x_{j}-\phi (p)_{j})\int _{0}^{1}\partial _{x_{j}}F(\xi \phi (p)+(1-\xi )x)d\xi \end{aligned}}}
  If one sets 
  
    
      
        x
        =
        ϕ
        (
        q
        )
      
    
    {\displaystyle x=\phi (q)}
   for 
  
    
      
        q
        ∈
        W
      
    
    {\displaystyle q\in W}
  , one obtains, inserting the definition of 
  
    
      
        F
      
    
    {\displaystyle F}
  :

  
    
      
        φ
        (
        q
        )
        =
        φ
        (
        p
        )
        +
        
          ∑
          
            j
            =
            1
          
          
            d
          
        
        (
        ϕ
        (
        q
        
          )
          
            j
          
        
        −
        ϕ
        (
        p
        
          )
          
            j
          
        
        )
        
          ∫
          
            0
          
          
            1
          
        
        
          ∂
          
            
              x
              
                j
              
            
          
        
        (
        φ
        ∘
        
          ϕ
          
            −
            1
          
        
        )
        (
        ξ
        ϕ
        (
        p
        )
        +
        (
        1
        −
        ξ
        )
        ϕ
        (
        q
        )
        )
        d
        ξ
      
    
    {\displaystyle \varphi (q)=\varphi (p)+\sum _{j=1}^{d}(\phi (q)_{j}-\phi (p)_{j})\int _{0}^{1}\partial _{x_{j}}(\varphi \circ \phi ^{-1})(\xi \phi (p)+(1-\xi )\phi (q))d\xi }
  Now we define the functions

  
    
      
        
          f
          
            j
          
        
        :
        W
        →
        
          R
        
        ,
        
          f
          
            j
          
        
        (
        q
        )
        :=
        
          ∫
          
            0
          
          
            1
          
        
        
          ∂
          
            
              x
              
                j
              
            
          
        
        (
        φ
        ∘
        
          ϕ
          
            −
            1
          
        
        )
        (
        ξ
        ϕ
        (
        p
        )
        +
        (
        1
        −
        ξ
        )
        ϕ
        (
        q
        )
        )
        d
        ξ
      
    
    {\displaystyle f_{j}:W\to \mathbb {R} ,f_{j}(q):=\int _{0}^{1}\partial _{x_{j}}(\varphi \circ \phi ^{-1})(\xi \phi (p)+(1-\xi )\phi (q))d\xi }
  These are contained in 
  
    
      
        
          
            
              C
            
          
          
            ∞
          
        
        (
        M
        )
      
    
    {\displaystyle {\mathcal {C}}^{\infty }(M)}
   since they are defined on 
  
    
      
        W
      
    
    {\displaystyle W}
   which is open, and further, if 
  
    
      
        (
        V
        ,
        θ
        )
        ∈
        {
        (
        
          O
          
            υ
          
        
        ,
        
          ϕ
          
            υ
          
        
        )
        
          |
        
        υ
        ∈
        Υ
        }
      
    
    {\displaystyle (V,\theta )\in \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}
  , then

  
    
      
        
          f
          
            j
          
        
        
          
            |
          
          
            V
            ∩
            W
          
        
        ∘
        θ
        
          
            |
          
          
            V
            ∩
            W
          
          
            −
            1
          
        
        =
        
          ∫
          
            0
          
          
            1
          
        
        
          ∂
          
            
              x
              
                j
              
            
          
        
        (
        φ
        ∘
        
          ϕ
          
            −
            1
          
        
        )
        (
        ξ
        ϕ
        
          
            |
          
          
            V
            ∩
            W
          
        
        ∘
        θ
        
          
            |
          
          
            V
            ∩
            W
          
          
            −
            1
          
        
        +
        (
        1
        −
        ξ
        )
        ϕ
        
          
            |
          
          
            V
            ∩
            W
          
        
        ∘
        θ
        
          
            |
          
          
            V
            ∩
            W
          
          
            −
            1
          
        
        )
        d
        ξ
      
    
    {\displaystyle f_{j}|_{V\cap W}\circ \theta |_{V\cap W}^{-1}=\int _{0}^{1}\partial _{x_{j}}(\varphi \circ \phi ^{-1})(\xi \phi |_{V\cap W}\circ \theta |_{V\cap W}^{-1}+(1-\xi )\phi |_{V\cap W}\circ \theta |_{V\cap W}^{-1})d\xi }
  , which is arbitrarily often differentiable by the Leibniz integral rule as the integral of a composition of arbitrarily often differentiable functions on a compact set.
Further, again denoting 
  
    
      
        ϕ
        =
        (
        
          ϕ
          
            1
          
        
        ,
        …
        ,
        
          ϕ
          
            d
          
        
        )
      
    
    {\displaystyle \phi =(\phi _{1},\ldots ,\phi _{d})}
  , the functions 
  
    
      
        
          ϕ
          
            k
          
        
      
    
    {\displaystyle \phi _{k}}
  , 
  
    
      
        k
        ∈
        {
        1
        ,
        …
        ,
        d
        }
      
    
    {\displaystyle k\in \{1,\ldots ,d\}}
   are contained in 
  
    
      
        
          
            
              C
            
          
          
            ∞
          
        
        (
        M
        )
      
    
    {\displaystyle {\mathcal {C}}^{\infty }(M)}
   due to lemma 2.3.
Since 
  
    
      
        φ
        
          
            |
          
          
            W
          
        
        ∈
        
          
            
              C
            
          
          
            ∞
          
        
        (
        M
        )
      
    
    {\displaystyle \varphi |_{W}\in {\mathcal {C}}^{\infty }(M)}
  , 
  
    
      
        
          
            V
          
          
            p
          
        
        (
        φ
        
          
            |
          
          
            W
          
        
        )
      
    
    {\displaystyle \mathbf {V} _{p}(\varphi |_{W})}
   is defined. We apply the rules (linearity and product rule) for tangent vectors and lemma 2.6 (we are allowed to do so because all the relevant functions are contained in 
  
    
      
        
          
            
              C
            
          
          
            ∞
          
        
        (
        M
        )
      
    
    {\displaystyle {\mathcal {C}}^{\infty }(M)}
  ), and obtain:

  
    
      
        
          
            
              
                
                  
                    V
                  
                  
                    p
                  
                
                (
                φ
                
                  
                    |
                  
                  
                    W
                  
                
                )
              
              
                
                =
                
                  
                    V
                  
                  
                    p
                  
                
                
                  (
                  
                    φ
                    (
                    p
                    )
                    +
                    
                      ∑
                      
                        j
                        =
                        1
                      
                      
                        d
                      
                    
                    (
                    
                      ϕ
                      
                        j
                      
                    
                    −
                    ϕ
                    (
                    p
                    
                      )
                      
                        j
                      
                    
                    )
                    
                      f
                      
                        j
                      
                    
                  
                  )
                
              
            
            
              
              
                
                =
                
                  ∑
                  
                    j
                    =
                    1
                  
                  
                    d
                  
                
                
                  (
                  
                    
                      ϕ
                      
                        j
                      
                    
                    (
                    p
                    )
                    
                      
                        V
                      
                      
                        p
                      
                    
                    (
                    
                      f
                      
                        j
                      
                    
                    )
                    +
                    
                      f
                      
                        j
                      
                    
                    (
                    p
                    )
                    
                      
                        V
                      
                      
                        p
                      
                    
                    (
                    
                      ϕ
                      
                        j
                      
                    
                    )
                    −
                    ϕ
                    (
                    p
                    
                      )
                      
                        j
                      
                    
                    
                      
                        V
                      
                      
                        p
                      
                    
                    (
                    
                      f
                      
                        j
                      
                    
                    )
                  
                  )
                
              
            
            
              
              
                
                =
                
                  ∑
                  
                    j
                    =
                    1
                  
                  
                    d
                  
                
                
                  f
                  
                    j
                  
                
                (
                p
                )
                
                  
                    V
                  
                  
                    p
                  
                
                (
                
                  ϕ
                  
                    j
                  
                
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\mathbf {V} _{p}(\varphi |_{W})&=\mathbf {V} _{p}\left(\varphi (p)+\sum _{j=1}^{d}(\phi _{j}-\phi (p)_{j})f_{j}\right)\\&=\sum _{j=1}^{d}\left(\phi _{j}(p)\mathbf {V} _{p}(f_{j})+f_{j}(p)\mathbf {V} _{p}(\phi _{j})-\phi (p)_{j}\mathbf {V} _{p}(f_{j})\right)\\&=\sum _{j=1}^{d}f_{j}(p)\mathbf {V} _{p}(\phi _{j})\end{aligned}}}
  , since due to our notation it's clear that 
  
    
      
        
          ϕ
          
            j
          
        
        (
        p
        )
        =
        ϕ
        (
        p
        
          )
          
            j
          
        
      
    
    {\displaystyle \phi _{j}(p)=\phi (p)_{j}}
  .
But

  
    
      
        
          
            
              
                
                  f
                  
                    j
                  
                
                (
                p
                )
              
              
                
                =
                
                  ∫
                  
                    0
                  
                  
                    1
                  
                
                
                  ∂
                  
                    
                      x
                      
                        j
                      
                    
                  
                
                (
                φ
                ∘
                
                  ϕ
                  
                    −
                    1
                  
                
                )
                (
                ξ
                ϕ
                (
                p
                )
                +
                (
                1
                −
                ξ
                )
                ϕ
                (
                p
                )
                )
                d
                ξ
              
            
            
              
              
                
                =
                
                  ∫
                  
                    0
                  
                  
                    1
                  
                
                
                  ∂
                  
                    
                      x
                      
                        j
                      
                    
                  
                
                (
                φ
                ∘
                
                  ϕ
                  
                    −
                    1
                  
                
                )
                (
                ϕ
                (
                p
                )
                )
                d
                ξ
              
            
            
              
              
                
                =
                
                  ∂
                  
                    
                      x
                      
                        j
                      
                    
                  
                
                (
                φ
                ∘
                
                  ϕ
                  
                    −
                    1
                  
                
                )
                (
                ϕ
                (
                p
                )
                )
              
            
            
              
              
                
                =
                
                  
                    (
                    
                      
                        ∂
                        
                          ∂
                          
                            ϕ
                            
                              j
                            
                          
                        
                      
                    
                    )
                  
                  
                    p
                  
                
                (
                φ
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}f_{j}(p)&=\int _{0}^{1}\partial _{x_{j}}(\varphi \circ \phi ^{-1})(\xi \phi (p)+(1-\xi )\phi (p))d\xi \\&=\int _{0}^{1}\partial _{x_{j}}(\varphi \circ \phi ^{-1})(\phi (p))d\xi \\&=\partial _{x_{j}}(\varphi \circ \phi ^{-1})(\phi (p))\\&=\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\varphi )\end{aligned}}}
  Thus we have successfully shown

  
    
      
        
          
            V
          
          
            p
          
        
        (
        φ
        
          
            |
          
          
            W
          
        
        )
        =
        
          ∑
          
            j
            =
            1
          
          
            d
          
        
        
          
            V
          
          
            p
          
        
        (
        
          ϕ
          
            j
          
        
        )
        
          
            (
            
              
                ∂
                
                  ∂
                  
                    ϕ
                    
                      j
                    
                  
                
              
            
            )
          
          
            p
          
        
        (
        φ
        )
      
    
    {\displaystyle \mathbf {V} _{p}(\varphi |_{W})=\sum _{j=1}^{d}\mathbf {V} _{p}(\phi _{j})\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\varphi )}
  But due to the definition of subtraction on 
  
    
      
        
          
            
              C
            
          
          
            ∞
          
        
        (
        M
        )
      
    
    {\displaystyle {\mathcal {C}}^{\infty }(M)}
  , due to lemma 2.6, and due to the fact that the constant zero function is a constant function:

  
    
      
        
          
            V
          
          
            p
          
        
        (
        φ
        
          
            |
          
          
            W
          
        
        −
        φ
        )
        =
        
          
            V
          
          
            p
          
        
        (
        0
        )
        =
        0
      
    
    {\displaystyle \mathbf {V} _{p}(\varphi |_{W}-\varphi )=\mathbf {V} _{p}(0)=0}
  Due to linearity of 
  
    
      
        
          
            V
          
          
            p
          
        
      
    
    {\displaystyle \mathbf {V} _{p}}
   follows 
  
    
      
        0
        =
        
          
            V
          
          
            p
          
        
        (
        φ
        
          
            |
          
          
            W
          
        
        )
        −
        
          
            V
          
          
            p
          
        
        (
        φ
        )
      
    
    {\displaystyle 0=\mathbf {V} _{p}(\varphi |_{W})-\mathbf {V} _{p}(\varphi )}
  , i. e. 
  
    
      
        
          
            V
          
          
            p
          
        
        (
        φ
        
          
            |
          
          
            W
          
        
        )
        =
        
          
            V
          
          
            p
          
        
        (
        φ
        )
      
    
    {\displaystyle \mathbf {V} _{p}(\varphi |_{W})=\mathbf {V} _{p}(\varphi )}
  . Now, inserting in the above equation gives the theorem.
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  
Together with theorem 2.5, this theorem shows that

  
    
      
        
          {
          
            
              
                (
                
                  
                    ∂
                    
                      ∂
                      
                        ϕ
                        
                          j
                        
                      
                    
                  
                
                )
              
              
                p
              
            
            
              
                |
              
            
            j
            ∈
            {
            1
            ,
            …
            ,
            d
            }
          
          }
        
      
    
    {\displaystyle \left\{\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}{\Bigg |}j\in \{1,\ldots ,d\}\right\}}
  is a basis of 
  
    
      
        
          T
          
            p
          
        
        M
      
    
    {\displaystyle T_{p}M}
  , because a basis is a linearly independent generating set. And since the dimension of a vector space was defined to be the number of elements in a basis, this implies that the dimension of 
  
    
      
        
          T
          
            p
          
        
        M
      
    
    {\displaystyle T_{p}M}
   is equal to 
  
    
      
        d
      
    
    {\displaystyle d}
  .


== Some bases of the cotangent space ==

Note that 
  
    
      
        (
        d
        
          ϕ
          
            j
          
        
        
          )
          
            p
          
        
      
    
    {\displaystyle (d\phi _{j})_{p}}
   is well-defined because of lemma 2.3.
Theorem 2.9: Let 
  
    
      
        M
      
    
    {\displaystyle M}
   be a 
  
    
      
        d
      
    
    {\displaystyle d}
  -dimensional manifold of class 
  
    
      
        
          
            
              C
            
          
          
            ∞
          
        
      
    
    {\displaystyle {\mathcal {C}}^{\infty }}
   and atlas 
  
    
      
        {
        (
        
          O
          
            υ
          
        
        ,
        
          ϕ
          
            υ
          
        
        )
        
          |
        
        υ
        ∈
        Υ
        }
      
    
    {\displaystyle \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}
  , let 
  
    
      
        (
        O
        ,
        ϕ
        )
        ∈
        {
        (
        
          O
          
            υ
          
        
        ,
        
          ϕ
          
            υ
          
        
        )
        
          |
        
        υ
        ∈
        Υ
        }
      
    
    {\displaystyle (O,\phi )\in \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}
   and let 
  
    
      
        p
        ∈
        O
      
    
    {\displaystyle p\in O}
  . For all 
  
    
      
        j
        ∈
        {
        1
        ,
        …
        ,
        d
        }
      
    
    {\displaystyle j\in \{1,\ldots ,d\}}
  , 
  
    
      
        (
        d
        
          ϕ
          
            j
          
        
        
          )
          
            p
          
        
      
    
    {\displaystyle (d\phi _{j})_{p}}
   is contained in 
  
    
      
        
          T
          
            p
          
        
        
          M
          
            ∗
          
        
      
    
    {\displaystyle T_{p}M^{*}}
  .
Proof:
By definition, 
  
    
      
        d
        
          ϕ
          
            k
          
        
      
    
    {\displaystyle d\phi _{k}}
   maps from 
  
    
      
        
          T
          
            p
          
        
        M
      
    
    {\displaystyle T_{p}M}
   to 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  . Thus, linearity is the only thing left to show. Indeed, for 
  
    
      
        
          
            V
          
          
            p
          
        
        ,
        
          
            W
          
          
            p
          
        
        ∈
        
          T
          
            p
          
        
        M
      
    
    {\displaystyle \mathbf {V} _{p},\mathbf {W} _{p}\in T_{p}M}
   and 
  
    
      
        b
        ∈
        
          R
        
      
    
    {\displaystyle b\in \mathbb {R} }
  , we have, since addition and scalar multiplication in 
  
    
      
        
          T
          
            p
          
        
        M
      
    
    {\displaystyle T_{p}M}
   are defined pointwise:

  
    
      
        
          
            
              
                (
                d
                
                  ϕ
                  
                    j
                  
                
                
                  )
                  
                    p
                  
                
                (
                
                  
                    V
                  
                  
                    p
                  
                
                +
                b
                
                  
                    W
                  
                  
                    p
                  
                
                )
              
              
                
                =
                (
                
                  
                    V
                  
                  
                    p
                  
                
                +
                b
                
                  
                    W
                  
                  
                    p
                  
                
                )
                (
                
                  ϕ
                  
                    k
                  
                
                )
              
            
            
              
              
                
                =
                
                  
                    V
                  
                  
                    p
                  
                
                (
                
                  ϕ
                  
                    k
                  
                
                )
                +
                b
                
                  
                    W
                  
                  
                    p
                  
                
                (
                
                  ϕ
                  
                    k
                  
                
                )
              
            
            
              
              
                
                =
                (
                d
                
                  ϕ
                  
                    j
                  
                
                
                  )
                  
                    p
                  
                
                (
                
                  
                    V
                  
                  
                    p
                  
                
                )
                +
                b
                (
                d
                
                  ϕ
                  
                    j
                  
                
                
                  )
                  
                    p
                  
                
                (
                
                  
                    W
                  
                  
                    p
                  
                
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}(d\phi _{j})_{p}(\mathbf {V} _{p}+b\mathbf {W} _{p})&=(\mathbf {V} _{p}+b\mathbf {W} _{p})(\phi _{k})\\&=\mathbf {V} _{p}(\phi _{k})+b\mathbf {W} _{p}(\phi _{k})\\&=(d\phi _{j})_{p}(\mathbf {V} _{p})+b(d\phi _{j})_{p}(\mathbf {W} _{p})\end{aligned}}}
  
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  Lemma 2.10: Let 
  
    
      
        M
      
    
    {\displaystyle M}
   be a 
  
    
      
        d
      
    
    {\displaystyle d}
  -dimensional manifold of class 
  
    
      
        
          
            
              C
            
          
          
            ∞
          
        
      
    
    {\displaystyle {\mathcal {C}}^{\infty }}
   and atlas 
  
    
      
        {
        (
        
          O
          
            υ
          
        
        ,
        
          ϕ
          
            υ
          
        
        )
        
          |
        
        υ
        ∈
        Υ
        }
      
    
    {\displaystyle \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}
  , let 
  
    
      
        (
        O
        ,
        ϕ
        )
        ∈
        {
        (
        
          O
          
            υ
          
        
        ,
        
          ϕ
          
            υ
          
        
        )
        
          |
        
        υ
        ∈
        Υ
        }
      
    
    {\displaystyle (O,\phi )\in \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}
   and let 
  
    
      
        p
        ∈
        O
      
    
    {\displaystyle p\in O}
  . For 
  
    
      
        j
        ,
        k
        ∈
        {
        1
        ,
        …
        ,
        d
        }
      
    
    {\displaystyle j,k\in \{1,\ldots ,d\}}
  , the following equation holds:

  
    
      
        (
        d
        
          ϕ
          
            j
          
        
        
          )
          
            p
          
        
        
          (
          
            
              (
              
                
                  ∂
                  
                    ∂
                    
                      ϕ
                      
                        k
                      
                    
                  
                
              
              )
            
            
              p
            
          
          )
        
        =
        
          
            {
            
              
                
                  1
                
                
                  k
                  =
                  j
                
              
              
                
                  0
                
                
                  k
                  ≠
                  j
                
              
            
            
          
        
      
    
    {\displaystyle (d\phi _{j})_{p}\left(\left({\frac {\partial }{\partial \phi _{k}}}\right)_{p}\right)={\begin{cases}1&k=j\\0&k\neq j\end{cases}}}
  Proof:
We have:

  
    
      
        (
        d
        
          ϕ
          
            j
          
        
        
          )
          
            p
          
        
        
          (
          
            
              (
              
                
                  ∂
                  
                    ∂
                    
                      ϕ
                      
                        k
                      
                    
                  
                
              
              )
            
            
              p
            
          
          )
        
        =
        
          
            (
            
              
                ∂
                
                  ∂
                  
                    ϕ
                    
                      k
                    
                  
                
              
            
            )
          
          
            p
          
        
        (
        
          ϕ
          
            j
          
        
        )
        
          
            =
            lemma 2.4
          
        
        
          
            {
            
              
                
                  1
                
                
                  k
                  =
                  j
                
              
              
                
                  0
                
                
                  k
                  ≠
                  j
                
              
            
            
          
        
      
    
    {\displaystyle (d\phi _{j})_{p}\left(\left({\frac {\partial }{\partial \phi _{k}}}\right)_{p}\right)=\left({\frac {\partial }{\partial \phi _{k}}}\right)_{p}(\phi _{j}){\overset {\text{lemma 2.4}}{=}}{\begin{cases}1&k=j\\0&k\neq j\end{cases}}}
  
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  Theorem 2.11: Let 
  
    
      
        M
      
    
    {\displaystyle M}
   be a 
  
    
      
        d
      
    
    {\displaystyle d}
  -dimensional manifold of class 
  
    
      
        
          
            
              C
            
          
          
            ∞
          
        
      
    
    {\displaystyle {\mathcal {C}}^{\infty }}
   and atlas 
  
    
      
        {
        (
        
          O
          
            υ
          
        
        ,
        
          ϕ
          
            υ
          
        
        )
        
          |
        
        υ
        ∈
        Υ
        }
      
    
    {\displaystyle \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}
  , let 
  
    
      
        (
        O
        ,
        ϕ
        )
        ∈
        {
        (
        
          O
          
            υ
          
        
        ,
        
          ϕ
          
            υ
          
        
        )
        
          |
        
        υ
        ∈
        Υ
        }
      
    
    {\displaystyle (O,\phi )\in \{(O_{\upsilon },\phi _{\upsilon })|\upsilon \in \Upsilon \}}
   and let 
  
    
      
        p
        ∈
        O
      
    
    {\displaystyle p\in O}
  . The cotangent vectors 
  
    
      
        (
        d
        
          ϕ
          
            j
          
        
        
          )
          
            p
          
        
        ,
        j
        ∈
        {
        1
        ,
        …
        ,
        d
        }
      
    
    {\displaystyle (d\phi _{j})_{p},j\in \{1,\ldots ,d\}}
   are linearly independent.
Proof:
Let 
  
    
      
        0
        =
        
          ∑
          
            j
            =
            1
          
          
            d
          
        
        
          a
          
            j
          
        
        (
        d
        
          ϕ
          
            j
          
        
        
          )
          
            p
          
        
      
    
    {\displaystyle 0=\sum _{j=1}^{d}a_{j}(d\phi _{j})_{p}}
  , where by 
  
    
      
        0
      
    
    {\displaystyle 0}
   we mean the zero of 
  
    
      
        
          T
          
            p
          
        
        
          M
          
            ∗
          
        
      
    
    {\displaystyle T_{p}M^{*}}
  . Then we have for all 
  
    
      
        k
        ∈
        {
        1
        ,
        …
        ,
        d
        }
      
    
    {\displaystyle k\in \{1,\ldots ,d\}}
  :

  
    
      
        0
        =
        
          ∑
          
            j
            =
            1
          
          
            d
          
        
        
          a
          
            j
          
        
        (
        d
        
          ϕ
          
            j
          
        
        
          )
          
            p
          
        
        
          (
          
            
              (
              
                
                  ∂
                  
                    ∂
                    
                      ϕ
                      
                        k
                      
                    
                  
                
              
              )
            
            
              p
            
          
          )
        
        
          
            =
            lemma 2.10
          
        
        
          a
          
            k
          
        
      
    
    {\displaystyle 0=\sum _{j=1}^{d}a_{j}(d\phi _{j})_{p}\left(\left({\frac {\partial }{\partial \phi _{k}}}\right)_{p}\right){\overset {\text{lemma 2.10}}{=}}a_{k}}
  
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  
Proof:
Let 
  
    
      
        
          α
          
            p
          
        
        ∈
        
          T
          
            p
          
        
        
          M
          
            ∗
          
        
      
    
    {\displaystyle \alpha _{p}\in T_{p}M^{*}}
   and 
  
    
      
        
          
            V
          
          
            p
          
        
        ∈
        
          T
          
            p
          
        
        M
      
    
    {\displaystyle \mathbf {V} _{p}\in T_{p}M}
  . Due to theorem 2.7, we have

  
    
      
        
          
            V
          
          
            p
          
        
        =
        
          ∑
          
            j
            =
            1
          
          
            d
          
        
        
          
            V
          
          
            p
          
        
        (
        
          ϕ
          
            j
          
        
        )
        
          
            (
            
              
                ∂
                
                  ∂
                  
                    ϕ
                    
                      j
                    
                  
                
              
            
            )
          
          
            p
          
        
      
    
    {\displaystyle \mathbf {V} _{p}=\sum _{j=1}^{d}\mathbf {V} _{p}(\phi _{j})\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}}
  Therefore, and due to the linearity of 
  
    
      
        
          α
          
            p
          
        
      
    
    {\displaystyle \alpha _{p}}
   (because 
  
    
      
        
          T
          
            p
          
        
        
          M
          
            ∗
          
        
      
    
    {\displaystyle T_{p}M^{*}}
   was the space of linear functions to 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  ):

  
    
      
        
          
            
              
                
                  α
                  
                    p
                  
                
                (
                
                  
                    V
                  
                  
                    p
                  
                
                )
              
              
                
                =
                
                  ∑
                  
                    j
                    =
                    1
                  
                  
                    d
                  
                
                
                  
                    V
                  
                  
                    p
                  
                
                (
                
                  ϕ
                  
                    j
                  
                
                )
                
                  α
                  
                    p
                  
                
                
                  (
                  
                    
                      (
                      
                        
                          ∂
                          
                            ∂
                            
                              ϕ
                              
                                j
                              
                            
                          
                        
                      
                      )
                    
                    
                      p
                    
                  
                  )
                
              
            
            
              
              
                
                =
                
                  ∑
                  
                    j
                    =
                    1
                  
                  
                    d
                  
                
                
                  α
                  
                    p
                  
                
                
                  (
                  
                    
                      (
                      
                        
                          ∂
                          
                            ∂
                            
                              ϕ
                              
                                j
                              
                            
                          
                        
                      
                      )
                    
                    
                      p
                    
                  
                  )
                
                (
                d
                
                  ϕ
                  
                    j
                  
                
                
                  )
                  
                    p
                  
                
                (
                
                  
                    V
                  
                  
                    p
                  
                
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\alpha _{p}(\mathbf {V} _{p})&=\sum _{j=1}^{d}\mathbf {V} _{p}(\phi _{j})\alpha _{p}\left(\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}\right)\\&=\sum _{j=1}^{d}\alpha _{p}\left(\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}\right)(d\phi _{j})_{p}(\mathbf {V} _{p})\end{aligned}}}
  Since 
  
    
      
        
          
            V
          
          
            p
          
        
        ∈
        
          T
          
            p
          
        
        M
      
    
    {\displaystyle \mathbf {V} _{p}\in T_{p}M}
   was arbitrary, the theorem is proven.
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  
From theorems 2.11 and 2.12 follows, as in the last subsection, that

  
    
      
        
          {
          
            (
            d
            
              ϕ
              
                j
              
            
            
              )
              
                p
              
            
            
              
                |
              
            
            j
            ∈
            {
            1
            ,
            …
            ,
            d
            }
          
          }
        
      
    
    {\displaystyle \left\{(d\phi _{j})_{p}{\big |}j\in \{1,\ldots ,d\}\right\}}
  is a basis for 
  
    
      
        
          T
          
            p
          
        
        
          M
          
            ∗
          
        
      
    
    {\displaystyle T_{p}M^{*}}
  , and that the dimension of 
  
    
      
        
          T
          
            p
          
        
        
          M
          
            ∗
          
        
      
    
    {\displaystyle T_{p}M^{*}}
   is equal to 
  
    
      
        d
      
    
    {\displaystyle d}
  , like the dimension of 
  
    
      
        
          T
          
            p
          
        
        M
      
    
    {\displaystyle T_{p}M}
  .


== Expressing elements of the tangent and cotangent spaces in different bases ==
If 
  
    
      
        M
      
    
    {\displaystyle M}
   is a manifold, 
  
    
      
        p
        ∈
        M
      
    
    {\displaystyle p\in M}
   and 
  
    
      
        (
        O
        ,
        ϕ
        )
        ,
        (
        U
        ,
        θ
        )
      
    
    {\displaystyle (O,\phi ),(U,\theta )}
   are two charts in 
  
    
      
        M
      
    
    {\displaystyle M}
  's atlas such that 
  
    
      
        p
        ∈
        O
      
    
    {\displaystyle p\in O}
   and 
  
    
      
        p
        ∈
        U
      
    
    {\displaystyle p\in U}
  . Then follows from the last two subsections, that

  
    
      
        
          {
          
            
              
                (
                
                  
                    ∂
                    
                      ∂
                      
                        ϕ
                        
                          j
                        
                      
                    
                  
                
                )
              
              
                p
              
            
            
              
                |
              
            
            j
            ∈
            {
            1
            ,
            …
            ,
            d
            }
          
          }
        
      
    
    {\displaystyle \left\{\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}{\Bigg |}j\in \{1,\ldots ,d\}\right\}}
   and 
  
    
      
        
          {
          
            
              
                (
                
                  
                    ∂
                    
                      ∂
                      
                        θ
                        
                          j
                        
                      
                    
                  
                
                )
              
              
                p
              
            
            
              
                |
              
            
            j
            ∈
            {
            1
            ,
            …
            ,
            d
            }
          
          }
        
      
    
    {\displaystyle \left\{\left({\frac {\partial }{\partial \theta _{j}}}\right)_{p}{\Bigg |}j\in \{1,\ldots ,d\}\right\}}
   are bases for 
  
    
      
        
          T
          
            p
          
        
        M
      
    
    {\displaystyle T_{p}M}
  , and

  
    
      
        
          {
          
            (
            d
            
              ϕ
              
                j
              
            
            
              )
              
                p
              
            
            
              
                |
              
            
            j
            ∈
            {
            1
            ,
            …
            ,
            d
            }
          
          }
        
      
    
    {\displaystyle \left\{(d\phi _{j})_{p}{\big |}j\in \{1,\ldots ,d\}\right\}}
   and 
  
    
      
        
          {
          
            (
            d
            
              θ
              
                j
              
            
            
              )
              
                p
              
            
            
              
                |
              
            
            j
            ∈
            {
            1
            ,
            …
            ,
            d
            }
          
          }
        
      
    
    {\displaystyle \left\{(d\theta _{j})_{p}{\big |}j\in \{1,\ldots ,d\}\right\}}
   are bases for 
  
    
      
        
          T
          
            p
          
        
        
          M
          
            ∗
          
        
      
    
    {\displaystyle T_{p}M^{*}}
  .One could now ask the questions:
If we have an element 
  
    
      
        
          
            V
          
          
            p
          
        
      
    
    {\displaystyle \mathbf {V} _{p}}
   in 
  
    
      
        
          T
          
            p
          
        
        M
      
    
    {\displaystyle T_{p}M}
   given by 
  
    
      
        
          
            V
          
          
            p
          
        
        =
        
          ∑
          
            j
            =
            1
          
          
            d
          
        
        
          a
          
            j
          
        
        
          
            (
            
              
                ∂
                
                  ∂
                  
                    ϕ
                    
                      j
                    
                  
                
              
            
            )
          
          
            p
          
        
      
    
    {\displaystyle \mathbf {V} _{p}=\sum _{j=1}^{d}a_{j}\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}}
  , then how can we represent 
  
    
      
        
          
            V
          
          
            p
          
        
      
    
    {\displaystyle \mathbf {V} _{p}}
   as linear combination of the basis 
  
    
      
        
          {
          
            
              
                (
                
                  
                    ∂
                    
                      ∂
                      
                        θ
                        
                          j
                        
                      
                    
                  
                
                )
              
              
                p
              
            
            
              
                |
              
            
            j
            ∈
            {
            1
            ,
            …
            ,
            d
            }
          
          }
        
      
    
    {\displaystyle \left\{\left({\frac {\partial }{\partial \theta _{j}}}\right)_{p}{\Bigg |}j\in \{1,\ldots ,d\}\right\}}
  ?
Or if we have an element 
  
    
      
        
          α
          
            p
          
        
      
    
    {\displaystyle \alpha _{p}}
   in 
  
    
      
        
          T
          
            p
          
        
        
          M
          
            ∗
          
        
      
    
    {\displaystyle T_{p}M^{*}}
   given by 
  
    
      
        
          α
          
            p
          
        
        =
        
          ∑
          
            j
            =
            1
          
          
            d
          
        
        
          a
          
            j
          
        
        (
        d
        
          ϕ
          
            j
          
        
        
          )
          
            p
          
        
      
    
    {\displaystyle \alpha _{p}=\sum _{j=1}^{d}a_{j}(d\phi _{j})_{p}}
  , then how can we represent 
  
    
      
        
          α
          
            p
          
        
      
    
    {\displaystyle \alpha _{p}}
   as linear combination of the basis 
  
    
      
        
          {
          
            (
            d
            
              θ
              
                j
              
            
            
              )
              
                p
              
            
            
              
                |
              
            
            j
            ∈
            {
            1
            ,
            …
            ,
            d
            }
          
          }
        
      
    
    {\displaystyle \left\{(d\theta _{j})_{p}{\big |}j\in \{1,\ldots ,d\}\right\}}
  ?
The following two theorems answer these questions:

Proof:
Due to theorem 2.7, we have for 
  
    
      
        j
        ∈
        {
        1
        ,
        …
        ,
        d
        }
      
    
    {\displaystyle j\in \{1,\ldots ,d\}}
  :

  
    
      
        
          
            (
            
              
                ∂
                
                  ∂
                  
                    ϕ
                    
                      j
                    
                  
                
              
            
            )
          
          
            p
          
        
        =
        
          ∑
          
            k
            =
            1
          
          
            d
          
        
        
          
            (
            
              
                ∂
                
                  ∂
                  
                    ϕ
                    
                      j
                    
                  
                
              
            
            )
          
          
            p
          
        
        (
        
          θ
          
            k
          
        
        )
        
          
            (
            
              
                ∂
                
                  ∂
                  
                    θ
                    
                      k
                    
                  
                
              
            
            )
          
          
            p
          
        
      
    
    {\displaystyle \left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}=\sum _{k=1}^{d}\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\theta _{k})\left({\frac {\partial }{\partial \theta _{k}}}\right)_{p}}
  From this follows:

  
    
      
        
          
            
              
                
                  
                    V
                  
                  
                    p
                  
                
              
              
                
                =
                
                  ∑
                  
                    j
                    =
                    1
                  
                  
                    d
                  
                
                
                  a
                  
                    j
                  
                
                
                  
                    (
                    
                      
                        ∂
                        
                          ∂
                          
                            ϕ
                            
                              j
                            
                          
                        
                      
                    
                    )
                  
                  
                    p
                  
                
              
            
            
              
              
                
                =
                
                  ∑
                  
                    j
                    =
                    1
                  
                  
                    d
                  
                
                
                  a
                  
                    j
                  
                
                
                  ∑
                  
                    k
                    =
                    1
                  
                  
                    d
                  
                
                
                  
                    (
                    
                      
                        ∂
                        
                          ∂
                          
                            ϕ
                            
                              j
                            
                          
                        
                      
                    
                    )
                  
                  
                    p
                  
                
                (
                
                  θ
                  
                    k
                  
                
                )
                
                  
                    (
                    
                      
                        ∂
                        
                          ∂
                          
                            θ
                            
                              k
                            
                          
                        
                      
                    
                    )
                  
                  
                    p
                  
                
              
            
            
              
              
                
                =
                
                  ∑
                  
                    k
                    =
                    1
                  
                  
                    d
                  
                
                
                  ∑
                  
                    j
                    =
                    1
                  
                  
                    d
                  
                
                
                  a
                  
                    j
                  
                
                
                  
                    (
                    
                      
                        ∂
                        
                          ∂
                          
                            ϕ
                            
                              j
                            
                          
                        
                      
                    
                    )
                  
                  
                    p
                  
                
                (
                
                  θ
                  
                    k
                  
                
                )
                
                  
                    (
                    
                      
                        ∂
                        
                          ∂
                          
                            θ
                            
                              k
                            
                          
                        
                      
                    
                    )
                  
                  
                    p
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\mathbf {V} _{p}&=\sum _{j=1}^{d}a_{j}\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}\\&=\sum _{j=1}^{d}a_{j}\sum _{k=1}^{d}\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\theta _{k})\left({\frac {\partial }{\partial \theta _{k}}}\right)_{p}\\&=\sum _{k=1}^{d}\sum _{j=1}^{d}a_{j}\left({\frac {\partial }{\partial \phi _{j}}}\right)_{p}(\theta _{k})\left({\frac {\partial }{\partial \theta _{k}}}\right)_{p}\end{aligned}}}
  
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  
Proof:
Due to theorem 2.12, we have for 
  
    
      
        j
        ∈
        {
        1
        ,
        …
        ,
        d
        }
      
    
    {\displaystyle j\in \{1,\ldots ,d\}}
  :

  
    
      
        (
        d
        
          ϕ
          
            j
          
        
        
          )
          
            p
          
        
        =
        
          ∑
          
            k
            =
            1
          
          
            d
          
        
        (
        d
        
          ϕ
          
            j
          
        
        
          )
          
            p
          
        
        
          (
          
            
              (
              
                
                  ∂
                  
                    ∂
                    
                      ϕ
                      
                        k
                      
                    
                  
                
              
              )
            
            
              p
            
          
          )
        
        (
        d
        
          θ
          
            k
          
        
        
          )
          
            p
          
        
      
    
    {\displaystyle (d\phi _{j})_{p}=\sum _{k=1}^{d}(d\phi _{j})_{p}\left(\left({\frac {\partial }{\partial \phi _{k}}}\right)_{p}\right)(d\theta _{k})_{p}}
  Thus we obtain:

  
    
      
        
          
            
              
                
                  α
                  
                    p
                  
                
              
              
                
                =
                
                  ∑
                  
                    j
                    =
                    1
                  
                  
                    d
                  
                
                
                  a
                  
                    j
                  
                
                (
                d
                
                  ϕ
                  
                    j
                  
                
                
                  )
                  
                    p
                  
                
              
            
            
              
              
                
                =
                
                  ∑
                  
                    j
                    =
                    1
                  
                  
                    d
                  
                
                
                  a
                  
                    j
                  
                
                
                  ∑
                  
                    k
                    =
                    1
                  
                  
                    d
                  
                
                (
                d
                
                  ϕ
                  
                    j
                  
                
                
                  )
                  
                    p
                  
                
                
                  (
                  
                    
                      (
                      
                        
                          ∂
                          
                            ∂
                            
                              ϕ
                              
                                k
                              
                            
                          
                        
                      
                      )
                    
                    
                      p
                    
                  
                  )
                
                (
                d
                
                  θ
                  
                    k
                  
                
                
                  )
                  
                    p
                  
                
              
            
            
              
              
                
                =
                
                  ∑
                  
                    k
                    =
                    1
                  
                  
                    d
                  
                
                
                  ∑
                  
                    j
                    =
                    1
                  
                  
                    d
                  
                
                
                  a
                  
                    j
                  
                
                (
                d
                
                  ϕ
                  
                    j
                  
                
                
                  )
                  
                    p
                  
                
                
                  (
                  
                    
                      (
                      
                        
                          ∂
                          
                            ∂
                            
                              ϕ
                              
                                k
                              
                            
                          
                        
                      
                      )
                    
                    
                      p
                    
                  
                  )
                
                (
                d
                
                  θ
                  
                    k
                  
                
                
                  )
                  
                    p
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\alpha _{p}&=\sum _{j=1}^{d}a_{j}(d\phi _{j})_{p}\\&=\sum _{j=1}^{d}a_{j}\sum _{k=1}^{d}(d\phi _{j})_{p}\left(\left({\frac {\partial }{\partial \phi _{k}}}\right)_{p}\right)(d\theta _{k})_{p}\\&=\sum _{k=1}^{d}\sum _{j=1}^{d}a_{j}(d\phi _{j})_{p}\left(\left({\frac {\partial }{\partial \phi _{k}}}\right)_{p}\right)(d\theta _{k})_{p}\end{aligned}}}
  
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  


== The pullback and the differentials ==
In this subsection, we will define the pullback and the differential. For the differential, we need three definitions, one for each of the following types of functions:

functions from a manifold to another manifold
functions from a manifold to 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
functions from an interval 
  
    
      
        I
        ⊆
        
          R
        
      
    
    {\displaystyle I\subseteq \mathbb {R} }
   to a manifold (i. e. curves)For the first of these, the differential of functions from a manifold to another manifold, we need to define what the pullback is:

Lemma 2.16: Let 
  
    
      
        M
      
    
    {\displaystyle M}
   be a 
  
    
      
        d
      
    
    {\displaystyle d}
  -dimensional and 
  
    
      
        N
      
    
    {\displaystyle N}
   be a 
  
    
      
        b
      
    
    {\displaystyle b}
  -dimensional manifold, let 
  
    
      
        k
        ∈
        
          
            N
          
          
            0
          
        
        ∪
        {
        ∞
        }
      
    
    {\displaystyle k\in \mathbb {N} _{0}\cup \{\infty \}}
   and let 
  
    
      
        ψ
        :
        M
        →
        N
      
    
    {\displaystyle \psi :M\to N}
   be differentiable of class 
  
    
      
        
          
            
              C
            
          
          
            k
          
        
      
    
    {\displaystyle {\mathcal {C}}^{k}}
  . Then 
  
    
      
        ψ
      
    
    {\displaystyle \psi }
   is continuous.
Proof:
We show that for an arbitrary 
  
    
      
        p
        ∈
        M
      
    
    {\displaystyle p\in M}
  , 
  
    
      
        ψ
      
    
    {\displaystyle \psi }
   is continuous on an open neighbourhood of 
  
    
      
        p
      
    
    {\displaystyle p}
  . There is a theorem in topology which states that from this follows continuity.
We choose 
  
    
      
        (
        O
        ,
        ϕ
        )
      
    
    {\displaystyle (O,\phi )}
   in the atlas of 
  
    
      
        M
      
    
    {\displaystyle M}
   such that 
  
    
      
        p
        ∈
        O
      
    
    {\displaystyle p\in O}
  , and 
  
    
      
        (
        U
        ,
        θ
        )
      
    
    {\displaystyle (U,\theta )}
   in the atlas of 
  
    
      
        N
      
    
    {\displaystyle N}
   such that 
  
    
      
        ψ
        (
        p
        )
        ∈
        U
      
    
    {\displaystyle \psi (p)\in U}
  . Due to the differentiability of 
  
    
      
        ψ
      
    
    {\displaystyle \psi }
  , the function

  
    
      
        θ
        ∘
        ψ
        ∘
        ϕ
        
          
            |
          
          
            ϕ
            (
            O
            ∩
            
              ψ
              
                −
                1
              
            
            (
            U
            )
            )
          
          
            −
            1
          
        
      
    
    {\displaystyle \theta \circ \psi \circ \phi |_{\phi (O\cap \psi ^{-1}(U))}^{-1}}
  is contained in 
  
    
      
        
          
            
              C
            
          
          
            k
          
        
        (
        
          
            R
          
          
            d
          
        
        ,
        
          
            R
          
          
            b
          
        
        )
      
    
    {\displaystyle {\mathcal {C}}^{k}(\mathbb {R} ^{d},\mathbb {R} ^{b})}
  , and therefore continuous. But 
  
    
      
        ϕ
      
    
    {\displaystyle \phi }
   and 
  
    
      
        θ
      
    
    {\displaystyle \theta }
   are charts and therefore homeomorphisms, and thus the function

  
    
      
        ψ
        
          
            |
          
          
            O
            ∩
            
              ψ
              
                −
                1
              
            
            (
            U
            )
          
        
        :
        O
        ∩
        
          ψ
          
            −
            1
          
        
        (
        U
        )
        →
        N
        ,
        ψ
        =
        
          θ
          
            −
            1
          
        
        ∘
        θ
        ∘
        ψ
        ∘
        ϕ
        
          
            |
          
          
            O
            ∩
            
              ψ
              
                −
                1
              
            
            (
            U
            )
          
          
            −
            1
          
        
        ∘
        ϕ
        
          
            |
          
          
            O
            ∩
            
              ψ
              
                −
                1
              
            
            (
            U
            )
          
        
      
    
    {\displaystyle \psi |_{O\cap \psi ^{-1}(U)}:O\cap \psi ^{-1}(U)\to N,\psi =\theta ^{-1}\circ \theta \circ \psi \circ \phi |_{O\cap \psi ^{-1}(U)}^{-1}\circ \phi |_{O\cap \psi ^{-1}(U)}}
  is continuous as the composition of continuous functions.
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  
Lemma 2.17: Let 
  
    
      
        M
        ,
        N
      
    
    {\displaystyle M,N}
   be two manifolds, let 
  
    
      
        ψ
        :
        M
        →
        N
      
    
    {\displaystyle \psi :M\to N}
   be differentiable of class 
  
    
      
        
          
            
              C
            
          
          
            k
          
        
      
    
    {\displaystyle {\mathcal {C}}^{k}}
  , and let 
  
    
      
        φ
        ∈
        
          
            
              C
            
          
          
            k
          
        
        (
        N
        )
      
    
    {\displaystyle \varphi \in {\mathcal {C}}^{k}(N)}
   be defined on the open set 
  
    
      
        U
        ⊆
        N
      
    
    {\displaystyle U\subseteq N}
  . In this case, the function 
  
    
      
        φ
        ∘
        
          
            |
          
          
            
              ψ
              
                −
                1
              
            
            (
            U
            )
          
        
      
    
    {\displaystyle \varphi \circ |_{\psi ^{-1}(U)}}
   is contained in 
  
    
      
        
          
            
              C
            
          
          
            k
          
        
        (
        M
        )
      
    
    {\displaystyle {\mathcal {C}}^{k}(M)}
  ; i. e. the pullback with respect to 
  
    
      
        ψ
      
    
    {\displaystyle \psi }
   really maps to 
  
    
      
        
          
            
              C
            
          
          
            k
          
        
        (
        M
        )
      
    
    {\displaystyle {\mathcal {C}}^{k}(M)}
  .
Proof:
Since 
  
    
      
        ψ
      
    
    {\displaystyle \psi }
   is continuous due to lemma 2.16, 
  
    
      
        
          ψ
          
            −
            1
          
        
        (
        U
        )
      
    
    {\displaystyle \psi ^{-1}(U)}
   is open in 
  
    
      
        M
      
    
    {\displaystyle M}
  . Thus 
  
    
      
        φ
        ∘
        ψ
        
          
            |
          
          
            
              ψ
              
                −
                1
              
            
            (
            U
            )
          
        
      
    
    {\displaystyle \varphi \circ \psi |_{\psi ^{-1}(U)}}
   is defined on an open set.
Let 
  
    
      
        (
        O
        ,
        ϕ
        )
      
    
    {\displaystyle (O,\phi )}
   be an arbitrary element of the atlas of 
  
    
      
        M
      
    
    {\displaystyle M}
   and let 
  
    
      
        x
        ∈
        ϕ
        (
        O
        )
      
    
    {\displaystyle x\in \phi (O)}
   be arbitrary. We choose 
  
    
      
        (
        V
        ,
        θ
        )
      
    
    {\displaystyle (V,\theta )}
   in the atlas of 
  
    
      
        N
      
    
    {\displaystyle N}
   such that 
  
    
      
        ψ
        (
        
          ϕ
          
            −
            1
          
        
        (
        x
        )
        )
        ∈
        V
      
    
    {\displaystyle \psi (\phi ^{-1}(x))\in V}
  . The function

  
    
      
        (
        φ
        ∘
        ψ
        
          
            |
          
          
            
              ψ
              
                −
                1
              
            
            (
            U
            )
          
        
        ∘
        ϕ
        
          
            |
          
          
            
              ψ
              
                −
                1
              
            
            (
            U
            )
            ∩
            O
          
          
            −
            1
          
        
        )
        
          
            |
          
          
            ϕ
            (
            
              ψ
              
                −
                1
              
            
            (
            U
            ∩
            V
            )
            ∩
            O
            )
          
        
        =
        φ
        
          
            |
          
          
            ψ
            (
            
              ψ
              
                −
                1
              
            
            (
            U
            ∩
            V
            )
            ∩
            O
            )
          
        
        ∘
        θ
        
          
            |
          
          
            ψ
            (
            
              ψ
              
                −
                1
              
            
            (
            U
            ∩
            V
            )
            ∩
            O
            )
          
          
            −
            1
          
        
        ∘
        θ
        
          
            |
          
          
            ψ
            (
            
              ψ
              
                −
                1
              
            
            (
            U
            ∩
            V
            )
            ∩
            O
            )
          
        
        ∘
        ψ
        
          
            |
          
          
            
              ψ
              
                −
                1
              
            
            (
            U
            ∩
            V
            )
            ∩
            O
          
        
        ∘
        ϕ
        
          
            |
          
          
            ϕ
            (
            
              ψ
              
                −
                1
              
            
            (
            U
            ∩
            V
            )
            ∩
            O
            )
          
          
            −
            1
          
        
      
    
    {\displaystyle (\varphi \circ \psi |_{\psi ^{-1}(U)}\circ \phi |_{\psi ^{-1}(U)\cap O}^{-1})|_{\phi (\psi ^{-1}(U\cap V)\cap O)}=\varphi |_{\psi (\psi ^{-1}(U\cap V)\cap O)}\circ \theta |_{\psi (\psi ^{-1}(U\cap V)\cap O)}^{-1}\circ \theta |_{\psi (\psi ^{-1}(U\cap V)\cap O)}\circ \psi |_{\psi ^{-1}(U\cap V)\cap O}\circ \phi |_{\phi (\psi ^{-1}(U\cap V)\cap O)}^{-1}}
  is 
  
    
      
        k
      
    
    {\displaystyle k}
  -times continuously differentiable (or continuous if 
  
    
      
        k
        =
        0
      
    
    {\displaystyle k=0}
  ) at 
  
    
      
        x
      
    
    {\displaystyle x}
   as the composition of two 
  
    
      
        k
      
    
    {\displaystyle k}
   times continuously differentiable (or continuous if 
  
    
      
        k
        =
        0
      
    
    {\displaystyle k=0}
  ) functions. Thus, the function

  
    
      
        φ
        ∘
        ψ
        
          
            |
          
          
            
              ψ
              
                −
                1
              
            
            (
            U
            )
          
        
        ∘
        ϕ
        
          
            |
          
          
            
              ψ
              
                −
                1
              
            
            (
            U
            )
            ∩
            O
          
          
            −
            1
          
        
      
    
    {\displaystyle \varphi \circ \psi |_{\psi ^{-1}(U)}\circ \phi |_{\psi ^{-1}(U)\cap O}^{-1}}
  is 
  
    
      
        k
      
    
    {\displaystyle k}
  -times continuously differentiable (or continuous if 
  
    
      
        k
        =
        0
      
    
    {\displaystyle k=0}
  ) at every point, and therefore contained in 
  
    
      
        
          
            
              C
            
          
          
            k
          
        
        (
        
          
            R
          
          
            d
          
        
        ,
        
          R
        
        )
      
    
    {\displaystyle {\mathcal {C}}^{k}(\mathbb {R} ^{d},\mathbb {R} )}
  .
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  

Theorem 2.19:
Let 
  
    
      
        M
        ,
        N
      
    
    {\displaystyle M,N}
   be two manifolds of class 
  
    
      
        
          
            
              C
            
          
          
            n
          
        
      
    
    {\displaystyle {\mathcal {C}}^{n}}
  , let 
  
    
      
        ψ
        :
        M
        →
        N
      
    
    {\displaystyle \psi :M\to N}
   be differentiable of class 
  
    
      
        
          
            
              C
            
          
          
            n
          
        
      
    
    {\displaystyle {\mathcal {C}}^{n}}
   and let 
  
    
      
        p
        ∈
        M
      
    
    {\displaystyle p\in M}
  . We have 
  
    
      
        
          
            V
          
          
            p
          
        
        ∘
        
          ψ
          
            ∗
          
        
        ∈
        
          T
          
            p
          
        
        N
      
    
    {\displaystyle \mathbf {V} _{p}\circ \psi ^{*}\in T_{p}N}
  ; i. e. the differential of 
  
    
      
        ψ
      
    
    {\displaystyle \psi }
   at 
  
    
      
        p
      
    
    {\displaystyle p}
   really maps to 
  
    
      
        
          T
          
            p
          
        
        N
      
    
    {\displaystyle T_{p}N}
  .
Proof:
Let 
  
    
      
        O
        ,
        U
        ⊆
        M
      
    
    {\displaystyle O,U\subseteq M}
   be open, 
  
    
      
        φ
        :
        O
        →
        
          R
        
        ,
        ϑ
        :
        U
        →
        
          R
        
        ∈
        
          
            
              C
            
          
          
            n
          
        
        (
        M
        )
      
    
    {\displaystyle \varphi :O\to \mathbb {R} ,\vartheta :U\to \mathbb {R} \in {\mathcal {C}}^{n}(M)}
   and 
  
    
      
        c
        ∈
        
          R
        
      
    
    {\displaystyle c\in \mathbb {R} }
   be arbitrary. In the proof of the following, we will use that for all open subsets 
  
    
      
        V
        ⊆
        O
      
    
    {\displaystyle V\subseteq O}
  , 
  
    
      
        
          
            V
          
          
            p
          
        
        (
        φ
        
          
            |
          
          
            V
          
        
        )
        =
        
          
            V
          
          
            p
          
        
        (
        φ
        )
      
    
    {\displaystyle \mathbf {V} _{p}(\varphi |_{V})=\mathbf {V} _{p}(\varphi )}
   (which follows from the linearity of 
  
    
      
        
          
            V
          
          
            p
          
        
      
    
    {\displaystyle \mathbf {V} _{p}}
  ).
1. We prove linearity.

  
    
      
        
          
            
              
                (
                
                  
                    V
                  
                  
                    p
                  
                
                ∘
                
                  ψ
                  
                    ∗
                  
                
                )
                (
                φ
                +
                c
                ϑ
                )
              
              
                
                =
                
                  
                    V
                  
                  
                    p
                  
                
                (
                
                  ψ
                  
                    ∗
                  
                
                (
                φ
                +
                c
                ϑ
                )
                )
              
            
            
              
              
                
                =
                
                  
                    V
                  
                  
                    p
                  
                
                (
                (
                φ
                +
                c
                ϑ
                )
                ∘
                ψ
                
                  
                    |
                  
                  
                    
                      ψ
                      
                        −
                        1
                      
                    
                    (
                    O
                    ∩
                    U
                    )
                  
                
                )
              
            
            
              
              
                
                =
                
                  
                    V
                  
                  
                    p
                  
                
                (
                φ
                
                  
                    |
                  
                  
                    O
                    ∩
                    U
                  
                
                ∘
                ψ
                
                  
                    |
                  
                  
                    
                      ψ
                      
                        −
                        1
                      
                    
                    (
                    O
                    ∩
                    U
                    )
                  
                
                +
                c
                ϑ
                
                  
                    |
                  
                  
                    O
                    ∩
                    U
                  
                
                ∘
                ψ
                
                  
                    |
                  
                  
                    
                      ψ
                      
                        −
                        1
                      
                    
                    (
                    O
                    ∩
                    U
                    )
                  
                
                )
              
            
            
              
              
                
                =
                
                  
                    V
                  
                  
                    p
                  
                
                (
                φ
                
                  
                    |
                  
                  
                    O
                    ∩
                    U
                  
                
                ∘
                ψ
                
                  
                    |
                  
                  
                    
                      ψ
                      
                        −
                        1
                      
                    
                    (
                    O
                    ∩
                    U
                    )
                  
                
                )
                +
                c
                
                  
                    V
                  
                  
                    p
                  
                
                (
                ϑ
                
                  
                    |
                  
                  
                    O
                    ∩
                    U
                  
                
                ∘
                ψ
                
                  
                    |
                  
                  
                    
                      ψ
                      
                        −
                        1
                      
                    
                    (
                    O
                    ∩
                    U
                    )
                  
                
                )
              
            
            
              
              
                
                =
                
                  
                    V
                  
                  
                    p
                  
                
                (
                
                  ψ
                  
                    ∗
                  
                
                (
                φ
                )
                )
                +
                c
                
                  
                    V
                  
                  
                    p
                  
                
                (
                
                  ψ
                  
                    ∗
                  
                
                (
                ϑ
                )
                )
              
            
            
              
              
                
                =
                (
                
                  
                    V
                  
                  
                    p
                  
                
                ∘
                
                  ψ
                  
                    ∗
                  
                
                )
                (
                φ
                )
                +
                c
                (
                
                  
                    V
                  
                  
                    p
                  
                
                ∘
                
                  ψ
                  
                    ∗
                  
                
                )
                (
                ϑ
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}(\mathbf {V} _{p}\circ \psi ^{*})(\varphi +c\vartheta )&=\mathbf {V} _{p}(\psi ^{*}(\varphi +c\vartheta ))\\&=\mathbf {V} _{p}((\varphi +c\vartheta )\circ \psi |_{\psi ^{-1}(O\cap U)})\\&=\mathbf {V} _{p}(\varphi |_{O\cap U}\circ \psi |_{\psi ^{-1}(O\cap U)}+c\vartheta |_{O\cap U}\circ \psi |_{\psi ^{-1}(O\cap U)})\\&=\mathbf {V} _{p}(\varphi |_{O\cap U}\circ \psi |_{\psi ^{-1}(O\cap U)})+c\mathbf {V} _{p}(\vartheta |_{O\cap U}\circ \psi |_{\psi ^{-1}(O\cap U)})\\&=\mathbf {V} _{p}(\psi ^{*}(\varphi ))+c\mathbf {V} _{p}(\psi ^{*}(\vartheta ))\\&=(\mathbf {V} _{p}\circ \psi ^{*})(\varphi )+c(\mathbf {V} _{p}\circ \psi ^{*})(\vartheta )\end{aligned}}}
  2. We prove the product rule.

  
    
      
        
          
            
              
                (
                
                  
                    V
                  
                  
                    p
                  
                
                ∘
                
                  ψ
                  
                    ∗
                  
                
                )
                (
                φ
                ϑ
                )
              
              
                
                =
                
                  
                    V
                  
                  
                    p
                  
                
                (
                
                  ψ
                  
                    ∗
                  
                
                (
                φ
                ϑ
                )
                )
              
            
            
              
              
                
                =
                
                  
                    V
                  
                  
                    p
                  
                
                (
                (
                φ
                
                  
                    |
                  
                  
                    O
                    ∩
                    U
                  
                
                ∘
                ψ
                
                  
                    |
                  
                  
                    
                      ψ
                      
                        −
                        1
                      
                    
                    (
                    O
                    ∩
                    U
                    )
                  
                
                )
                (
                ϑ
                
                  
                    |
                  
                  
                    O
                    ∩
                    U
                  
                
                ∘
                ψ
                
                  
                    |
                  
                  
                    
                      ψ
                      
                        −
                        1
                      
                    
                    (
                    O
                    ∩
                    U
                    )
                  
                
                )
                )
              
            
            
              
              
                
                =
                (
                φ
                
                  
                    |
                  
                  
                    O
                    ∩
                    U
                  
                
                ∘
                ψ
                
                  
                    |
                  
                  
                    
                      ψ
                      
                        −
                        1
                      
                    
                    (
                    O
                    ∩
                    U
                    )
                  
                
                )
                (
                p
                )
                
                  
                    V
                  
                  
                    p
                  
                
                (
                ϑ
                
                  
                    |
                  
                  
                    O
                    ∩
                    U
                  
                
                ∘
                ψ
                
                  
                    |
                  
                  
                    
                      ψ
                      
                        −
                        1
                      
                    
                    (
                    O
                    ∩
                    U
                    )
                  
                
                )
                +
                (
                ϑ
                
                  
                    |
                  
                  
                    O
                    ∩
                    U
                  
                
                ∘
                ψ
                
                  
                    |
                  
                  
                    
                      ψ
                      
                        −
                        1
                      
                    
                    (
                    O
                    ∩
                    U
                    )
                  
                
                )
                (
                p
                )
                
                  
                    V
                  
                  
                    p
                  
                
                (
                φ
                
                  
                    |
                  
                  
                    O
                    ∩
                    U
                  
                
                ∘
                ψ
                
                  
                    |
                  
                  
                    
                      ψ
                      
                        −
                        1
                      
                    
                    (
                    O
                    ∩
                    U
                    )
                  
                
                )
              
            
            
              
              
                
                =
                φ
                (
                ψ
                (
                p
                )
                )
                
                  
                    V
                  
                  
                    p
                  
                
                (
                
                  ψ
                  
                    ∗
                  
                
                ϑ
                )
                +
                ϑ
                (
                ψ
                (
                p
                )
                )
                
                  
                    V
                  
                  
                    p
                  
                
                (
                
                  ψ
                  
                    ∗
                  
                
                φ
                )
              
            
            
              
              
                
                =
                φ
                (
                ψ
                (
                p
                )
                )
                (
                
                  
                    V
                  
                  
                    p
                  
                
                ∘
                
                  ψ
                  
                    ∗
                  
                
                )
                (
                ϑ
                )
                +
                ϑ
                (
                ψ
                (
                p
                )
                )
                (
                
                  
                    V
                  
                  
                    p
                  
                
                ∘
                
                  ψ
                  
                    ∗
                  
                
                )
                (
                φ
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}(\mathbf {V} _{p}\circ \psi ^{*})(\varphi \vartheta )&=\mathbf {V} _{p}(\psi ^{*}(\varphi \vartheta ))\\&=\mathbf {V} _{p}((\varphi |_{O\cap U}\circ \psi |_{\psi ^{-1}(O\cap U)})(\vartheta |_{O\cap U}\circ \psi |_{\psi ^{-1}(O\cap U)}))\\&=(\varphi |_{O\cap U}\circ \psi |_{\psi ^{-1}(O\cap U)})(p)\mathbf {V} _{p}(\vartheta |_{O\cap U}\circ \psi |_{\psi ^{-1}(O\cap U)})+(\vartheta |_{O\cap U}\circ \psi |_{\psi ^{-1}(O\cap U)})(p)\mathbf {V} _{p}(\varphi |_{O\cap U}\circ \psi |_{\psi ^{-1}(O\cap U)})\\&=\varphi (\psi (p))\mathbf {V} _{p}(\psi ^{*}\vartheta )+\vartheta (\psi (p))\mathbf {V} _{p}(\psi ^{*}\varphi )\\&=\varphi (\psi (p))(\mathbf {V} _{p}\circ \psi ^{*})(\vartheta )+\vartheta (\psi (p))(\mathbf {V} _{p}\circ \psi ^{*})(\varphi )\end{aligned}}}
  
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  

Theorem 2.22: Let 
  
    
      
        M
      
    
    {\displaystyle M}
   be a manifold of class 
  
    
      
        
          
            
              C
            
          
          
            n
          
        
      
    
    {\displaystyle {\mathcal {C}}^{n}}
  , 
  
    
      
        n
        ≥
        1
      
    
    {\displaystyle n\geq 1}
  , let 
  
    
      
        I
        ⊆
        
          R
        
      
    
    {\displaystyle I\subseteq \mathbb {R} }
   be an interval, let 
  
    
      
        y
        ∈
        I
      
    
    {\displaystyle y\in I}
   and let 
  
    
      
        γ
        :
        I
        →
        M
      
    
    {\displaystyle \gamma :I\to M}
   be a differentiable curve of class 
  
    
      
        
          
            
              C
            
          
          
            n
          
        
      
    
    {\displaystyle {\mathcal {C}}^{n}}
  . Then 
  
    
      
        φ
        ∘
        γ
      
    
    {\displaystyle \varphi \circ \gamma }
   is contained in 
  
    
      
        
          
            
              C
            
          
          
            n
          
        
        (
        
          R
        
        ,
        
          R
        
        )
      
    
    {\displaystyle {\mathcal {C}}^{n}(\mathbb {R} ,\mathbb {R} )}
   for every 
  
    
      
        φ
        ∈
        
          
            
              C
            
          
          
            n
          
        
        (
        M
        )
      
    
    {\displaystyle \varphi \in {\mathcal {C}}^{n}(M)}
   and 
  
    
      
        
          γ
          
            y
          
          ′
        
      
    
    {\displaystyle \gamma '_{y}}
   is a tangent vector of 
  
    
      
        M
      
    
    {\displaystyle M}
   at 
  
    
      
        γ
        (
        y
        )
      
    
    {\displaystyle \gamma (y)}
  .
Proof:
1. We show 
  
    
      
        ∀
        φ
        ∈
        
          
            
              C
            
          
          
            n
          
        
        (
        M
        )
        :
        ∘
        γ
        ∈
        
          
            
              C
            
          
          
            n
          
        
        (
        
          R
        
        ,
        
          R
        
        )
      
    
    {\displaystyle \forall \varphi \in {\mathcal {C}}^{n}(M):\circ \gamma \in {\mathcal {C}}^{n}(\mathbb {R} ,\mathbb {R} )}
  
Let 
  
    
      
        x
        ∈
        I
      
    
    {\displaystyle x\in I}
   be arbitrary, and let 
  
    
      
        U
      
    
    {\displaystyle U}
   be the set where 
  
    
      
        φ
      
    
    {\displaystyle \varphi }
   is defined (
  
    
      
        U
      
    
    {\displaystyle U}
   is open by the definition of 
  
    
      
        
          
            
              C
            
          
          
            n
          
        
        (
        M
        )
      
    
    {\displaystyle {\mathcal {C}}^{n}(M)}
   functions. We choose 
  
    
      
        (
        O
        ,
        ϕ
        )
      
    
    {\displaystyle (O,\phi )}
   in the atlas of 
  
    
      
        M
      
    
    {\displaystyle M}
   such that 
  
    
      
        γ
        (
        x
        )
        ∈
        O
      
    
    {\displaystyle \gamma (x)\in O}
  . Then the function

  
    
      
        (
        φ
        ∘
        γ
        )
        
          
            |
          
          
            
              γ
              
                −
                1
              
            
            (
            O
            ∩
            U
            )
            ∩
            I
          
        
        =
        φ
        ∘
        
          ϕ
          
            −
            1
          
        
        ∘
        ϕ
        ∘
        γ
        
          
            |
          
          
            
              γ
              
                −
                1
              
            
            (
            O
            ∩
            U
            )
            ∩
            I
          
        
      
    
    {\displaystyle (\varphi \circ \gamma )|_{\gamma ^{-1}(O\cap U)\cap I}=\varphi \circ \phi ^{-1}\circ \phi \circ \gamma |_{\gamma ^{-1}(O\cap U)\cap I}}
  is contained in 
  
    
      
        
          
            
              C
            
          
          
            n
          
        
        (
        
          R
        
        ,
        
          R
        
        )
      
    
    {\displaystyle {\mathcal {C}}^{n}(\mathbb {R} ,\mathbb {R} )}
   as the composition of two 
  
    
      
        n
      
    
    {\displaystyle n}
   times continuously differentiable (or continuous if 
  
    
      
        n
        =
        0
      
    
    {\displaystyle n=0}
  ) functions.
Thus, 
  
    
      
        φ
        ∘
        γ
      
    
    {\displaystyle \varphi \circ \gamma }
   is 
  
    
      
        n
      
    
    {\displaystyle n}
   times continuously differentiable (or continuous if 
  
    
      
        n
        =
        0
      
    
    {\displaystyle n=0}
  ) at every point, and hence 
  
    
      
        n
      
    
    {\displaystyle n}
   times continuously differentiable (or continuous if 
  
    
      
        n
        =
        0
      
    
    {\displaystyle n=0}
  ).
2. We show that 
  
    
      
        
          γ
          
            y
          
          ′
        
        ∈
        
          T
          
            γ
            (
            y
            )
          
        
        M
      
    
    {\displaystyle \gamma '_{y}\in T_{\gamma (y)}M}
   in three steps:
Let 
  
    
      
        φ
        ,
        ϑ
        ∈
        
          
            
              C
            
          
          
            n
          
        
        (
        M
        )
      
    
    {\displaystyle \varphi ,\vartheta \in {\mathcal {C}}^{n}(M)}
   and 
  
    
      
        c
        ∈
        
          R
        
      
    
    {\displaystyle c\in \mathbb {R} }
  .
2.1 We show linearity.
We have:

  
    
      
        
          
            
              
                
                  γ
                  
                    y
                  
                  ′
                
                (
                φ
                +
                c
                ϑ
                )
              
              
                
                =
                (
                (
                φ
                +
                c
                ϑ
                )
                ∘
                γ
                
                  )
                  ′
                
                (
                y
                )
              
            
            
              
              
                
                =
                (
                φ
                ∘
                γ
                +
                c
                ϑ
                ∘
                γ
                
                  )
                  ′
                
                (
                y
                )
              
            
            
              
              
                
                =
                (
                φ
                ∘
                γ
                
                  )
                  ′
                
                (
                y
                )
                +
                c
                (
                ϑ
                ∘
                γ
                
                  )
                  ′
                
                (
                y
                )
              
            
            
              
              
                
                =
                
                  γ
                  
                    y
                  
                  ′
                
                (
                φ
                )
                +
                c
                
                  γ
                  
                    y
                  
                  ′
                
                (
                ϑ
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\gamma '_{y}(\varphi +c\vartheta )&=((\varphi +c\vartheta )\circ \gamma )'(y)\\&=(\varphi \circ \gamma +c\vartheta \circ \gamma )'(y)\\&=(\varphi \circ \gamma )'(y)+c(\vartheta \circ \gamma )'(y)\\&=\gamma '_{y}(\varphi )+c\gamma '_{y}(\vartheta )\end{aligned}}}
  2.2 We prove the product rule.

  
    
      
        
          
            
              
                
                  γ
                  
                    y
                  
                  ′
                
                (
                φ
                ϑ
                )
              
              
                
                =
                (
                (
                φ
                ϑ
                )
                ∘
                γ
                
                  )
                  ′
                
                (
                y
                )
              
            
            
              
              
                
                =
                (
                (
                φ
                ∘
                γ
                )
                (
                ϑ
                ∘
                γ
                )
                
                  )
                  ′
                
                (
                y
                )
              
            
            
              
              
                
                =
                (
                φ
                ∘
                γ
                )
                (
                y
                )
                (
                ϑ
                ∘
                γ
                
                  )
                  ′
                
                (
                y
                )
                +
                (
                ϑ
                ∘
                γ
                )
                (
                y
                )
                (
                φ
                ∘
                γ
                
                  )
                  ′
                
                (
                y
                )
              
            
            
              
              
                
                =
                φ
                (
                γ
                (
                y
                )
                )
                
                  γ
                  
                    y
                  
                  ′
                
                (
                ϑ
                )
                +
                ϑ
                (
                γ
                (
                y
                )
                )
                
                  γ
                  
                    y
                  
                  ′
                
                (
                φ
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\gamma '_{y}(\varphi \vartheta )&=((\varphi \vartheta )\circ \gamma )'(y)\\&=((\varphi \circ \gamma )(\vartheta \circ \gamma ))'(y)\\&=(\varphi \circ \gamma )(y)(\vartheta \circ \gamma )'(y)+(\vartheta \circ \gamma )(y)(\varphi \circ \gamma )'(y)\\&=\varphi (\gamma (y))\gamma '_{y}(\vartheta )+\vartheta (\gamma (y))\gamma '_{y}(\varphi )\end{aligned}}}
  2.3 It follows from the definition of 
  
    
      
        
          γ
          
            y
          
          ′
        
      
    
    {\displaystyle \gamma '_{y}}
   that 
  
    
      
        
          γ
          
            y
          
          ′
        
        (
        φ
        )
      
    
    {\displaystyle \gamma '_{y}(\varphi )}
   is equal to zero if 
  
    
      
        φ
      
    
    {\displaystyle \varphi }
   is not defined at 
  
    
      
        γ
        (
        y
        )
      
    
    {\displaystyle \gamma (y)}
  .
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  


== Linearity of the differential for Ck(M), product, quotient and chain rules ==
In this subsection, we will first prove linearity and product rule for functions from a manifold to 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  .

Proof:
1. We show that 
  
    
      
        φ
        +
        c
        ϑ
        ∈
        
          
            
              C
            
          
          
            k
          
        
        (
        M
        )
      
    
    {\displaystyle \varphi +c\vartheta \in {\mathcal {C}}^{k}(M)}
  .
Let 
  
    
      
        U
      
    
    {\displaystyle U}
   be the (open as intersection of two open sets) set on which 
  
    
      
        φ
        +
        c
        ϑ
      
    
    {\displaystyle \varphi +c\vartheta }
   is defined, and let 
  
    
      
        (
        O
        ,
        ϕ
        )
      
    
    {\displaystyle (O,\phi )}
   be contained in the atlas of 
  
    
      
        M
      
    
    {\displaystyle M}
  . The function

  
    
      
        (
        φ
        +
        c
        ϑ
        )
        
          
            |
          
          
            O
            ∩
            U
          
        
        ∘
        ϕ
        
          
            |
          
          
            O
            ∩
            U
          
          
            −
            1
          
        
        =
        φ
        
          
            |
          
          
            O
            ∩
            U
          
        
        ∘
        ϕ
        
          
            |
          
          
            O
            ∩
            U
          
          
            −
            1
          
        
        +
        c
        ϑ
        
          
            |
          
          
            O
            ∩
            U
          
        
        ∘
        ϕ
        
          
            |
          
          
            O
            ∩
            U
          
          
            −
            1
          
        
      
    
    {\displaystyle (\varphi +c\vartheta )|_{O\cap U}\circ \phi |_{O\cap U}^{-1}=\varphi |_{O\cap U}\circ \phi |_{O\cap U}^{-1}+c\vartheta |_{O\cap U}\circ \phi |_{O\cap U}^{-1}}
  is contained in 
  
    
      
        
          
            
              C
            
          
          
            n
          
        
        (
        
          
            R
          
          
            d
          
        
        ,
        
          R
        
        )
      
    
    {\displaystyle {\mathcal {C}}^{n}(\mathbb {R} ^{d},\mathbb {R} )}
   as the linear combination of two 
  
    
      
        
          
            
              C
            
          
          
            n
          
        
        (
        
          
            R
          
          
            d
          
        
        ,
        
          R
        
        )
      
    
    {\displaystyle {\mathcal {C}}^{n}(\mathbb {R} ^{d},\mathbb {R} )}
   functions.
2. We show that 
  
    
      
        d
        (
        φ
        +
        c
        ϑ
        )
        =
        d
        φ
        +
        c
        d
        ϑ
      
    
    {\displaystyle d(\varphi +c\vartheta )=d\varphi +cd\vartheta }
  .
For all 
  
    
      
        p
        ∈
        M
      
    
    {\displaystyle p\in M}
   and 
  
    
      
        
          
            V
          
          
            p
          
        
        ∈
        
          T
          
            p
          
        
        M
      
    
    {\displaystyle \mathbf {V} _{p}\in T_{p}M}
  , we have:

  
    
      
        d
        (
        φ
        +
        c
        ϑ
        
          )
          
            p
          
        
        (
        
          
            V
          
          
            p
          
        
        )
        =
        
          
            V
          
          
            p
          
        
        (
        φ
        +
        c
        ϑ
        )
        =
        
          
            V
          
          
            p
          
        
        (
        φ
        )
        +
        c
        
          
            V
          
          
            p
          
        
        (
        ϑ
        )
        =
        d
        
          φ
          
            p
          
        
        (
        
          
            V
          
          
            p
          
        
        )
        +
        c
        d
        
          ϑ
          
            p
          
        
        (
        
          
            V
          
          
            p
          
        
        )
      
    
    {\displaystyle d(\varphi +c\vartheta )_{p}(\mathbf {V} _{p})=\mathbf {V} _{p}(\varphi +c\vartheta )=\mathbf {V} _{p}(\varphi )+c\mathbf {V} _{p}(\vartheta )=d\varphi _{p}(\mathbf {V} _{p})+cd\vartheta _{p}(\mathbf {V} _{p})}
  
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  Remark 2.24: This also shows that for all 
  
    
      
        φ
        ∈
        
          
            
              C
            
          
          
            n
          
        
        (
        M
        )
      
    
    {\displaystyle \varphi \in {\mathcal {C}}^{n}(M)}
  , 
  
    
      
        d
        
          φ
          
            p
          
        
        ∈
        
          T
          
            p
          
        
        
          M
          
            ∗
          
        
      
    
    {\displaystyle d\varphi _{p}\in T_{p}M^{*}}
  .

Proof:
1. We show that 
  
    
      
        φ
        ϑ
        ∈
        
          
            
              C
            
          
          
            k
          
        
        (
        M
        )
      
    
    {\displaystyle \varphi \vartheta \in {\mathcal {C}}^{k}(M)}
  .
Let 
  
    
      
        U
      
    
    {\displaystyle U}
   be the (open as intersection of two open sets) set on which 
  
    
      
        φ
        ϑ
      
    
    {\displaystyle \varphi \vartheta }
   is defined, and let 
  
    
      
        (
        O
        ,
        ϕ
        )
      
    
    {\displaystyle (O,\phi )}
   be contained in the atlas of 
  
    
      
        M
      
    
    {\displaystyle M}
  . The function

  
    
      
        (
        φ
        ϑ
        )
        
          
            |
          
          
            O
            ∩
            U
          
        
        ∘
        ϕ
        
          
            |
          
          
            O
            ∩
            U
          
          
            −
            1
          
        
        =
        φ
        
          
            |
          
          
            O
            ∩
            U
          
        
        ∘
        ϕ
        
          
            |
          
          
            O
            ∩
            U
          
          
            −
            1
          
        
        ϑ
        
          
            |
          
          
            O
            ∩
            U
          
        
        ∘
        ϕ
        
          
            |
          
          
            O
            ∩
            U
          
          
            −
            1
          
        
      
    
    {\displaystyle (\varphi \vartheta )|_{O\cap U}\circ \phi |_{O\cap U}^{-1}=\varphi |_{O\cap U}\circ \phi |_{O\cap U}^{-1}\vartheta |_{O\cap U}\circ \phi |_{O\cap U}^{-1}}
  is contained in 
  
    
      
        
          
            
              C
            
          
          
            n
          
        
        (
        
          
            R
          
          
            d
          
        
        ,
        
          R
        
        )
      
    
    {\displaystyle {\mathcal {C}}^{n}(\mathbb {R} ^{d},\mathbb {R} )}
   as the product of two 
  
    
      
        
          
            
              C
            
          
          
            n
          
        
        (
        
          
            R
          
          
            d
          
        
        ,
        
          R
        
        )
      
    
    {\displaystyle {\mathcal {C}}^{n}(\mathbb {R} ^{d},\mathbb {R} )}
   functions.
2. We show that 
  
    
      
        d
        (
        φ
        ϑ
        )
        =
        φ
        d
        ϑ
        +
        ϑ
        d
        φ
      
    
    {\displaystyle d(\varphi \vartheta )=\varphi d\vartheta +\vartheta d\varphi }
  .
For all 
  
    
      
        p
        ∈
        M
      
    
    {\displaystyle p\in M}
   and 
  
    
      
        
          
            V
          
          
            p
          
        
        ∈
        
          T
          
            p
          
        
        M
      
    
    {\displaystyle \mathbf {V} _{p}\in T_{p}M}
  , we have:

  
    
      
        d
        (
        φ
        ϑ
        
          )
          
            p
          
        
        (
        
          
            V
          
          
            p
          
        
        )
        =
        
          
            V
          
          
            p
          
        
        (
        φ
        ϑ
        )
        =
        φ
        (
        p
        )
        
          
            V
          
          
            p
          
        
        (
        ϑ
        )
        +
        ϑ
        (
        p
        )
        
          
            V
          
          
            p
          
        
        (
        φ
        )
        =
        φ
        (
        p
        )
        d
        
          ϑ
          
            p
          
        
        +
        ϑ
        (
        p
        )
        d
        
          φ
          
            p
          
        
      
    
    {\displaystyle d(\varphi \vartheta )_{p}(\mathbf {V} _{p})=\mathbf {V} _{p}(\varphi \vartheta )=\varphi (p)\mathbf {V} _{p}(\vartheta )+\vartheta (p)\mathbf {V} _{p}(\varphi )=\varphi (p)d\vartheta _{p}+\vartheta (p)d\varphi _{p}}
  
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  
Proof:
1. We show that 
  
    
      
        
          
            φ
            ϑ
          
        
        ∈
        
          
            
              C
            
          
          
            n
          
        
        (
        M
        )
      
    
    {\displaystyle {\frac {\varphi }{\vartheta }}\in {\mathcal {C}}^{n}(M)}
  :
Let 
  
    
      
        U
      
    
    {\displaystyle U}
   be the (open as the intersection of two open set) set on which 
  
    
      
        
          
            φ
            ϑ
          
        
      
    
    {\displaystyle {\frac {\varphi }{\vartheta }}}
   is defined, and let 
  
    
      
        (
        O
        ,
        ϕ
        )
      
    
    {\displaystyle (O,\phi )}
   be in the atlas of 
  
    
      
        M
      
    
    {\displaystyle M}
   such that 
  
    
      
        O
        ∩
        U
        ≠
        ∅
      
    
    {\displaystyle O\cap U\neq \emptyset }
  . The function

  
    
      
        
          
            φ
            ϑ
          
        
        
          
            
              |
            
          
          
            O
            ∩
            U
          
        
        ∘
        ϕ
        
          
            |
          
          
            O
            ∩
            U
          
          
            −
            1
          
        
        =
        
          
            
              φ
              
                
                  |
                
                
                  O
                  ∩
                  U
                
              
              ∘
              ϕ
              
                
                  |
                
                
                  O
                  ∩
                  U
                
                
                  −
                  1
                
              
            
            
              ϑ
              
                
                  |
                
                
                  O
                  ∩
                  U
                
              
              ∘
              ϕ
              
                
                  |
                
                
                  O
                  ∩
                  U
                
                
                  −
                  1
                
              
            
          
        
      
    
    {\displaystyle {\frac {\varphi }{\vartheta }}{\big |}_{O\cap U}\circ \phi |_{O\cap U}^{-1}={\frac {\varphi |_{O\cap U}\circ \phi |_{O\cap U}^{-1}}{\vartheta |_{O\cap U}\circ \phi |_{O\cap U}^{-1}}}}
  is contained in 
  
    
      
        
          
            
              C
            
          
          
            n
          
        
        (
        
          
            R
          
          
            d
          
        
        ,
        
          R
        
        )
      
    
    {\displaystyle {\mathcal {C}}^{n}(\mathbb {R} ^{d},\mathbb {R} )}
   as the quotient of two 
  
    
      
        
          
            
              C
            
          
          
            n
          
        
        (
        
          
            R
          
          
            d
          
        
        ,
        
          R
        
        )
      
    
    {\displaystyle {\mathcal {C}}^{n}(\mathbb {R} ^{d},\mathbb {R} )}
   from which the function in the denominator vanishes nowhere.
2. We show that 
  
    
      
        d
        
          (
          
            
              φ
              ϑ
            
          
          )
        
        =
        
          
            
              ϑ
              d
              φ
              −
              φ
              d
              ϑ
            
            
              ϑ
              
                2
              
            
          
        
      
    
    {\displaystyle d\left({\frac {\varphi }{\vartheta }}\right)={\frac {\vartheta d\varphi -\varphi d\vartheta }{\vartheta ^{2}}}}
  :
Choosing 
  
    
      
        φ
      
    
    {\displaystyle \varphi }
   as the constant one function, we obtain from 1. that the function 
  
    
      
        
          
            1
            ϑ
          
        
      
    
    {\displaystyle {\frac {1}{\vartheta }}}
   is in 
  
    
      
        
          
            
              C
            
          
          
            n
          
        
        (
        M
        )
      
    
    {\displaystyle {\mathcal {C}}^{n}(M)}
  . Hence follows from the product rule:

  
    
      
        0
        =
        d
        
          (
          
            ϑ
            
              
                1
                ϑ
              
            
          
          )
        
        =
        ϑ
        d
        
          (
          
            
              1
              ϑ
            
          
          )
        
        +
        
          
            1
            ϑ
          
        
        d
        ϑ
      
    
    {\displaystyle 0=d\left(\vartheta {\frac {1}{\vartheta }}\right)=\vartheta d\left({\frac {1}{\vartheta }}\right)+{\frac {1}{\vartheta }}d\vartheta }
  which, through equivalent transformations, can be transformed to

  
    
      
        d
        
          (
          
            
              1
              ϑ
            
          
          )
        
        =
        −
        
          
            
              d
              ϑ
            
            
              ϑ
              
                2
              
            
          
        
      
    
    {\displaystyle d\left({\frac {1}{\vartheta }}\right)=-{\frac {d\vartheta }{\vartheta ^{2}}}}
  From this and from the product rule we obtain:

  
    
      
        d
        
          (
          
            φ
            
              
                1
                ϑ
              
            
          
          )
        
        =
        
          
            1
            ϑ
          
        
        d
        φ
        −
        
          
            
              φ
              d
              ϑ
            
            
              ϑ
              
                2
              
            
          
        
        =
        
          
            
              ϑ
              d
              φ
              −
              φ
              d
              ϑ
            
            
              ϑ
              
                2
              
            
          
        
      
    
    {\displaystyle d\left(\varphi {\frac {1}{\vartheta }}\right)={\frac {1}{\vartheta }}d\varphi -{\frac {\varphi d\vartheta }{\vartheta ^{2}}}={\frac {\vartheta d\varphi -\varphi d\vartheta }{\vartheta ^{2}}}}
  
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  
Proof:
1. We already know that 
  
    
      
        φ
        ∘
        ψ
        =
        
          ψ
          
            ∗
          
        
        φ
      
    
    {\displaystyle \varphi \circ \psi =\psi ^{*}\varphi }
   is differentiable of class 
  
    
      
        
          
            
              C
            
          
          
            n
          
        
      
    
    {\displaystyle {\mathcal {C}}^{n}}
  ; this is what lemma 2.17 says.
2. We prove that 
  
    
      
        d
        (
        
          ψ
          
            ∗
          
        
        φ
        
          )
          
            p
          
        
        =
        d
        
          φ
          
            ψ
            (
            p
            )
          
        
        ∘
        d
        
          ψ
          
            p
          
        
      
    
    {\displaystyle d(\psi ^{*}\varphi )_{p}=d\varphi _{\psi (p)}\circ d\psi _{p}}
  .
Let 
  
    
      
        
          
            V
          
          
            p
          
        
        ∈
        
          T
          
            p
          
        
        M
      
    
    {\displaystyle \mathbf {V} _{p}\in T_{p}M}
  . Then we have:

  
    
      
        
          
            
              
                (
                d
                
                  φ
                  
                    ψ
                    (
                    p
                    )
                  
                
                ∘
                d
                
                  ψ
                  
                    p
                  
                
                )
                (
                
                  
                    V
                  
                  
                    p
                  
                
                )
              
              
                
                =
                d
                
                  φ
                  
                    ψ
                    (
                    p
                    )
                  
                
                (
                d
                
                  ψ
                  
                    p
                  
                
                (
                
                  
                    V
                  
                  
                    p
                  
                
                )
                )
              
            
            
              
              
                
                =
                d
                
                  φ
                  
                    ψ
                    (
                    p
                    )
                  
                
                (
                
                  
                    V
                  
                  
                    p
                  
                
                ∘
                ψ
                )
              
            
            
              
              
                
                =
                (
                
                  
                    V
                  
                  
                    p
                  
                
                ∘
                
                  ψ
                  
                    ∗
                  
                
                )
                (
                φ
                )
              
            
            
              
              
                
                =
                
                  
                    V
                  
                  
                    p
                  
                
                (
                
                  ψ
                  
                    ∗
                  
                
                (
                φ
                )
                )
              
            
            
              
              
                
                =
                
                  
                    V
                  
                  
                    p
                  
                
                (
                φ
                ∘
                ψ
                )
              
            
            
              
              
                
                =
                d
                (
                φ
                ∘
                ψ
                
                  )
                  
                    p
                  
                
                (
                
                  
                    V
                  
                  
                    p
                  
                
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}(d\varphi _{\psi (p)}\circ d\psi _{p})(\mathbf {V} _{p})&=d\varphi _{\psi (p)}(d\psi _{p}(\mathbf {V} _{p}))\\&=d\varphi _{\psi (p)}(\mathbf {V} _{p}\circ \psi )\\&=(\mathbf {V} _{p}\circ \psi ^{*})(\varphi )\\&=\mathbf {V} _{p}(\psi ^{*}(\varphi ))\\&=\mathbf {V} _{p}(\varphi \circ \psi )\\&=d(\varphi \circ \psi )_{p}(\mathbf {V} _{p})\end{aligned}}}
  
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  Now, let's go on to proving the chain rule for functions from manifolds to manifolds. But to do so, we first need another theorem about the pullback.
Theorem 2.28:
Let 
  
    
      
        M
        ,
        N
        ,
        K
      
    
    {\displaystyle M,N,K}
   be three manifolds, and let 
  
    
      
        ψ
        :
        M
        →
        N
      
    
    {\displaystyle \psi :M\to N}
   and 
  
    
      
        χ
        :
        N
        →
        K
      
    
    {\displaystyle \chi :N\to K}
   be two functions differentiable of class 
  
    
      
        
          
            
              C
            
          
          
            k
          
        
      
    
    {\displaystyle {\mathcal {C}}^{k}}
  . Then

  
    
      
        (
        χ
        ∘
        ψ
        
          )
          
            ∗
          
        
        =
        
          ψ
          
            ∗
          
        
        ∘
        
          χ
          
            ∗
          
        
      
    
    {\displaystyle (\chi \circ \psi )^{*}=\psi ^{*}\circ \chi ^{*}}
  Proof:
Let 
  
    
      
        φ
        ∈
        
          
            
              C
            
          
          
            k
          
        
        (
        K
        )
      
    
    {\displaystyle \varphi \in {\mathcal {C}}^{k}(K)}
  . Then we have:

  
    
      
        
          
            
              
                (
                χ
                ∘
                ψ
                
                  )
                  
                    ∗
                  
                
                (
                φ
                )
              
              
                
                =
                φ
                ∘
                (
                χ
                ∘
                ψ
                )
              
            
            
              
              
                
                =
                (
                φ
                ∘
                χ
                )
                ∘
                ψ
              
            
            
              
              
                
                =
                
                  χ
                  
                    ∗
                  
                
                (
                φ
                )
                ∘
                ψ
              
            
            
              
              
                
                =
                
                  ψ
                  
                    ∗
                  
                
                (
                
                  χ
                  
                    ∗
                  
                
                (
                φ
                )
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}(\chi \circ \psi )^{*}(\varphi )&=\varphi \circ (\chi \circ \psi )\\&=(\varphi \circ \chi )\circ \psi \\&=\chi ^{*}(\varphi )\circ \psi \\&=\psi ^{*}(\chi ^{*}(\varphi ))\end{aligned}}}
  
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  
Proof:
1. We prove that 
  
    
      
        χ
        ∘
        ψ
      
    
    {\displaystyle \chi \circ \psi }
   is differentiable of class 
  
    
      
        
          
            
              C
            
          
          
            k
          
        
      
    
    {\displaystyle {\mathcal {C}}^{k}}
  .
Let 
  
    
      
        (
        O
        ,
        ϕ
        )
      
    
    {\displaystyle (O,\phi )}
   be contained in the atlas of 
  
    
      
        M
      
    
    {\displaystyle M}
   and let 
  
    
      
        (
        U
        ,
        θ
        )
      
    
    {\displaystyle (U,\theta )}
   be contained in the atlas of 
  
    
      
        K
      
    
    {\displaystyle K}
   such that 
  
    
      
        O
        ∩
        
          ψ
          
            −
            1
          
        
        (
        
          χ
          
            −
            1
          
        
        (
        U
        )
        )
        ≠
        ∅
      
    
    {\displaystyle O\cap \psi ^{-1}(\chi ^{-1}(U))\neq \emptyset }
  , and let 
  
    
      
        x
        ∈
        
          ϕ
          
            −
            1
          
        
        (
        O
        )
        ∩
        
          ψ
          
            −
            1
          
        
        (
        
          χ
          
            −
            1
          
        
        (
        U
        )
        )
      
    
    {\displaystyle x\in \phi ^{-1}(O)\cap \psi ^{-1}(\chi ^{-1}(U))}
   be arbitrary. We choose 
  
    
      
        (
        V
        ,
        η
        )
      
    
    {\displaystyle (V,\eta )}
   in the atlas of 
  
    
      
        N
      
    
    {\displaystyle N}
   such that 
  
    
      
        ψ
        (
        ϕ
        (
        x
        )
        )
        ∈
        V
      
    
    {\displaystyle \psi (\phi (x))\in V}
  .
We have 
  
    
      
        ψ
        (
        ϕ
        (
        x
        )
        )
        ∈
        V
        ∩
        
          χ
          
            −
            1
          
        
        (
        U
        )
      
    
    {\displaystyle \psi (\phi (x))\in V\cap \chi ^{-1}(U)}
  ; indeed, 
  
    
      
        ψ
        (
        ϕ
        (
        x
        )
        )
        ∈
        V
      
    
    {\displaystyle \psi (\phi (x))\in V}
   due to the choice of 
  
    
      
        (
        V
        ,
        ϕ
        )
      
    
    {\displaystyle (V,\phi )}
   and 
  
    
      
        ψ
        (
        ϕ
        (
        x
        )
        )
        ∈
        
          χ
          
            −
            1
          
        
        (
        U
        )
      
    
    {\displaystyle \psi (\phi (x))\in \chi ^{-1}(U)}
   because 
  
    
      
        ϕ
        (
        x
        )
        ∈
        
          ψ
          
            −
            1
          
        
        (
        
          χ
          
            −
            1
          
        
        (
        U
        )
        )
      
    
    {\displaystyle \phi (x)\in \psi ^{-1}(\chi ^{-1}(U))}
  . Further, we choose 
  
    
      
        W
        :=
        O
        ∩
        
          ψ
          
            −
            1
          
        
        (
        V
        ∩
        
          χ
          
            −
            1
          
        
        (
        U
        )
        )
      
    
    {\displaystyle W:=O\cap \psi ^{-1}(V\cap \chi ^{-1}(U))}
  . Then the function

  
    
      
        
          θ
          
            −
            1
          
        
        ∘
        (
        χ
        ∘
        ψ
        )
        ∘
        ϕ
        
          
            |
          
          
            W
          
          
            −
            1
          
        
        =
        
          θ
          
            −
            1
          
        
        ∘
        χ
        ∘
        
          η
          
            −
            1
          
        
        ∘
        η
        ∘
        ψ
        
          
            |
          
          
            W
          
        
        ∘
        ϕ
        
          
            |
          
          
            W
          
          
            −
            1
          
        
      
    
    {\displaystyle \theta ^{-1}\circ (\chi \circ \psi )\circ \phi |_{W}^{-1}=\theta ^{-1}\circ \chi \circ \eta ^{-1}\circ \eta \circ \psi |_{W}\circ \phi |_{W}^{-1}}
  is contained in 
  
    
      
        
          
            
              C
            
          
          
            n
          
        
        (
        
          
            R
          
          
            d
          
        
        ,
        
          
            R
          
          
            d
          
        
        )
      
    
    {\displaystyle {\mathcal {C}}^{n}(\mathbb {R} ^{d},\mathbb {R} ^{d})}
   as the composition of two 
  
    
      
        
          
            
              C
            
          
          
            n
          
        
        (
        
          
            R
          
          
            d
          
        
        ,
        
          
            R
          
          
            d
          
        
        )
      
    
    {\displaystyle {\mathcal {C}}^{n}(\mathbb {R} ^{d},\mathbb {R} ^{d})}
   functions.
Thus, 
  
    
      
        
          θ
          
            −
            1
          
        
        ∘
        (
        χ
        ∘
        ψ
        )
        ∘
        ϕ
        
          
            |
          
          
            O
            ∩
            
              ψ
              
                −
                1
              
            
            (
            
              χ
              
                −
                1
              
            
            (
            U
            )
            )
          
          
            −
            1
          
        
      
    
    {\displaystyle \theta ^{-1}\circ (\chi \circ \psi )\circ \phi |_{O\cap \psi ^{-1}(\chi ^{-1}(U))}^{-1}}
   is 
  
    
      
        n
      
    
    {\displaystyle n}
   times continuously differentiable (or continuous if 
  
    
      
        n
        =
        0
      
    
    {\displaystyle n=0}
  ) at every point, and thus 
  
    
      
        n
      
    
    {\displaystyle n}
   times continuously differentiable (or continuous if 
  
    
      
        n
        =
        0
      
    
    {\displaystyle n=0}
  ).
2. We prove that 
  
    
      
        ∀
        p
        ∈
        M
        :
        d
        (
        χ
        ∘
        ψ
        
          )
          
            p
          
        
        =
        d
        
          χ
          
            ψ
            (
            p
            )
          
        
        ∘
        d
        
          ψ
          
            p
          
        
      
    
    {\displaystyle \forall p\in M:d(\chi \circ \psi )_{p}=d\chi _{\psi (p)}\circ d\psi _{p}}
  .
For all 
  
    
      
        p
        ∈
        M
      
    
    {\displaystyle p\in M}
   and 
  
    
      
        
          
            V
          
          
            p
          
        
        ∈
        
          T
          
            p
          
        
        M
      
    
    {\displaystyle \mathbf {V} _{p}\in T_{p}M}
  , we have:

  
    
      
        
          
            
              
                (
                d
                
                  χ
                  
                    ψ
                    (
                    p
                    )
                  
                
                ∘
                d
                
                  ψ
                  
                    p
                  
                
                )
                (
                
                  
                    V
                  
                  
                    p
                  
                
                )
              
              
                
                =
                d
                
                  χ
                  
                    ψ
                    (
                    p
                    )
                  
                
                (
                d
                
                  ψ
                  
                    p
                  
                
                (
                
                  
                    V
                  
                  
                    p
                  
                
                )
                )
              
            
            
              
              
                
                =
                d
                
                  χ
                  
                    ψ
                    (
                    p
                    )
                  
                
                (
                
                  
                    V
                  
                  
                    p
                  
                
                ∘
                
                  ψ
                  
                    ∗
                  
                
                )
              
            
            
              
              
                
                =
                
                  
                    V
                  
                  
                    p
                  
                
                ∘
                
                  ψ
                  
                    ∗
                  
                
                ∘
                
                  χ
                  
                    ∗
                  
                
              
            
            
              
              
                
                
                  
                    =
                    theorem 2.26
                  
                
                
                  
                    V
                  
                  
                    p
                  
                
                ∘
                (
                χ
                ∘
                ψ
                
                  )
                  
                    ∗
                  
                
              
            
            
              
              
                
                =
                d
                (
                χ
                ∘
                ψ
                
                  )
                  
                    p
                  
                
                (
                
                  
                    V
                  
                  
                    p
                  
                
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}(d\chi _{\psi (p)}\circ d\psi _{p})(\mathbf {V} _{p})&=d\chi _{\psi (p)}(d\psi _{p}(\mathbf {V} _{p}))\\&=d\chi _{\psi (p)}(\mathbf {V} _{p}\circ \psi ^{*})\\&=\mathbf {V} _{p}\circ \psi ^{*}\circ \chi ^{*}\\&{\overset {\text{theorem 2.26}}{=}}\mathbf {V} _{p}\circ (\chi \circ \psi )^{*}\\&=d(\chi \circ \psi )_{p}(\mathbf {V} _{p})\end{aligned}}}
  
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  
Proof:
1. Among another thing, theorem 2.22 states that 
  
    
      
        φ
        ∘
        γ
      
    
    {\displaystyle \varphi \circ \gamma }
   is contained in 
  
    
      
        
          
            
              C
            
          
          
            n
          
        
        (
        
          R
        
        ,
        
          R
        
        )
      
    
    {\displaystyle {\mathcal {C}}^{n}(\mathbb {R} ,\mathbb {R} )}
  .
2. We show that 
  
    
      
        (
        φ
        ∘
        γ
        
          )
          ′
        
        (
        y
        )
        =
        d
        
          φ
          
            γ
            (
            y
            )
          
        
        (
        
          γ
          
            y
          
          ′
        
        )
      
    
    {\displaystyle (\varphi \circ \gamma )'(y)=d\varphi _{\gamma (y)}(\gamma '_{y})}
  :

  
    
      
        
          
            
              
                d
                
                  φ
                  
                    γ
                    (
                    y
                    )
                  
                
                (
                
                  γ
                  
                    y
                  
                  ′
                
                )
              
              
                
                =
                
                  γ
                  
                    y
                  
                  ′
                
                (
                φ
                )
              
            
            
              
              
                
                =
                (
                φ
                ∘
                γ
                
                  )
                  ′
                
                (
                y
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}d\varphi _{\gamma (y)}(\gamma '_{y})&=\gamma '_{y}(\varphi )\\&=(\varphi \circ \gamma )'(y)\end{aligned}}}
  
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  


== Intuition behind the tangent space ==
In this section, we want to prove that what we defined as the tangent space is isomorphic to a space whose elements are in analogy to tangent vectors to, say, tangent vectors of a function 
  
    
      
        f
        :
        
          R
        
        →
        
          R
        
      
    
    {\displaystyle f:\mathbb {R} \to \mathbb {R} }
  .
We start by proving the following lemma from linear algebra:

Proof:
We only prove that 
  
    
      
        T
      
    
    {\displaystyle T}
   is a vector space isomorphism; that 
  
    
      
        S
      
    
    {\displaystyle S}
   and 
  
    
      
        L
      
    
    {\displaystyle L}
   are also vector space isomorphisms will follow in exactly the same way.
From 
  
    
      
        L
        ∘
        S
        ∘
        T
        =
        
          
            Id
          
          
            
              V
            
          
        
      
    
    {\displaystyle L\circ S\circ T={\text{Id}}_{\mathbf {V} }}
   and 
  
    
      
        T
        ∘
        L
        ∘
        S
        =
        
          
            Id
          
          
            
              W
            
          
        
      
    
    {\displaystyle T\circ L\circ S={\text{Id}}_{\mathbf {W} }}
   follows that 
  
    
      
        L
        ∘
        S
      
    
    {\displaystyle L\circ S}
   is the inverse function of 
  
    
      
        T
      
    
    {\displaystyle T}
  .
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  


== Sources ==
Torres del Castillo, Gerardo (2012). Differentiable Manifolds. Boston: Birkhäuser. ISBN 978-0-8176-8271-2.