[<< wikibooks] Differentiable Manifolds/Orientation
Proof: The given family of functions defines an atlas on 
  
    
      
        ∂
        M
      
    
    {\displaystyle \partial M}
  , so that the proposition will be proven once it is demonstrated that the requirement regarding the positivity of the determinant is satisfied.
Indeed, let 
  
    
      
        α
        ,
        β
        ∈
        A
      
    
    {\displaystyle \alpha ,\beta \in A}
  . Then

  
    
      
        
          π
          
            2
            ,
            …
            ,
            n
          
        
        ∘
        
          φ
          
            α
          
        
        ↾
        ∂
        M
        ∘
        (
        
          π
          
            2
            ,
            …
            ,
            n
          
        
        ∘
        
          φ
          
            β
          
        
        ↾
        ∂
        M
        
          )
          
            −
            1
          
        
        =
        
          π
          
            2
            ,
            …
            ,
            n
          
        
        ∘
        (
        
          φ
          
            α
          
        
        ∘
        
          φ
          
            β
          
          
            −
            1
          
        
        )
        ∘
        (
        
          π
          
            2
            ,
            …
            ,
            n
          
        
        ↾
        {
        (
        
          x
          
            1
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
        )
        ∈
        
          
            R
          
          
            n
          
        
        
          |
        
        
          x
          
            1
          
        
        =
        0
        }
        
          )
          
            −
            1
          
        
      
    
    {\displaystyle \pi _{2,\ldots ,n}\circ \varphi _{\alpha }\upharpoonright \partial M\circ (\pi _{2,\ldots ,n}\circ \varphi _{\beta }\upharpoonright \partial M)^{-1}=\pi _{2,\ldots ,n}\circ (\varphi _{\alpha }\circ \varphi _{\beta }^{-1})\circ (\pi _{2,\ldots ,n}\upharpoonright \{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}=0\})^{-1}}
  .Since 
  
    
      
        
          φ
          
            α
          
        
        ∘
        
          φ
          
            β
          
          
            −
            1
          
        
      
    
    {\displaystyle \varphi _{\alpha }\circ \varphi _{\beta }^{-1}}
   maps the set 
  
    
      
        {
        (
        
          x
          
            1
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
        )
        ∈
        
          
            R
          
          
            n
          
        
        
          |
        
        
          x
          
            1
          
        
        =
        0
        }
      
    
    {\displaystyle \{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}=0\}}
   to itself, the first row of 
  
    
      
        D
        (
        
          φ
          
            α
          
        
        ∘
        
          φ
          
            β
          
          
            −
            1
          
        
        )
      
    
    {\displaystyle D(\varphi _{\alpha }\circ \varphi _{\beta }^{-1})}
   is zero so long as 
  
    
      
        x
        ∈
        {
        (
        
          x
          
            1
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
        )
        ∈
        
          
            R
          
          
            n
          
        
        
          |
        
        
          x
          
            1
          
        
        =
        0
        }
      
    
    {\displaystyle x\in \{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}=0\}}
  , except the very first entry. Yet the very last entry must be non-negative so long as 
  
    
      
        x
        ∈
        {
        (
        
          x
          
            1
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
        )
        ∈
        
          
            R
          
          
            n
          
        
        
          |
        
        
          x
          
            1
          
        
        =
        0
        }
      
    
    {\displaystyle x\in \{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}=0\}}
  , since otherwise the fundamental theorem of calculus and the continuity of the derivative would imply that for a 
  
    
      
        y
        :=
        (
        
          y
          
            1
          
        
        ,
        …
        ,
        
          y
          
            n
          
        
        )
        ∈
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle y:=(y_{1},\ldots ,y_{n})\in \mathbb {R} ^{n}}
   such that 
  
    
      
        
          y
          
            n
          
        
        >
        0
      
    
    {\displaystyle y_{n}>0}
   was sufficiently small, 
  
    
      
        
          φ
          
            α
          
        
        ∘
        
          φ
          
            β
          
          
            −
            1
          
        
        (
        y
        )
      
    
    {\displaystyle \varphi _{\alpha }\circ \varphi _{\beta }^{-1}(y)}
   would be contained within

  
    
      
        {
        (
        
          x
          
            1
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
        )
        ∈
        
          
            R
          
          
            n
          
        
        
          |
        
        
          x
          
            n
          
        
        <
        0
        }
      
    
    {\displaystyle \{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{n}<0\}}
  ,contrary to the definition of charts that contain a piece of the boundary. Hence, upon carrying out a Leibniz expansion of 
  
    
      
        D
        (
        
          φ
          
            α
          
        
        ∘
        
          φ
          
            β
          
          
            −
            1
          
        
        )
      
    
    {\displaystyle D(\varphi _{\alpha }\circ \varphi _{\beta }^{-1})}
   along the first row, we obtain that the matrix obtained from 
  
    
      
        D
        (
        
          φ
          
            α
          
        
        ∘
        
          φ
          
            β
          
          
            −
            1
          
        
        )
      
    
    {\displaystyle D(\varphi _{\alpha }\circ \varphi _{\beta }^{-1})}
   by removing the first row and the first column has positive determinant. Yet the definitions of the respective partial derivatives as limits show that this matrix is exactly the matrix

  
    
      
        D
        (
        
          π
          
            2
            ,
            …
            ,
            n
          
        
        ∘
        (
        
          φ
          
            α
          
        
        ∘
        
          φ
          
            β
          
          
            −
            1
          
        
        )
        ∘
        (
        
          π
          
            2
            ,
            …
            ,
            n
          
        
        ↾
        {
        (
        
          x
          
            1
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
        )
        ∈
        
          
            R
          
          
            n
          
        
        
          |
        
        
          x
          
            1
          
        
        =
        0
        }
        
          )
          
            −
            1
          
        
        )
      
    
    {\displaystyle D(\pi _{2,\ldots ,n}\circ (\varphi _{\alpha }\circ \varphi _{\beta }^{-1})\circ (\pi _{2,\ldots ,n}\upharpoonright \{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}|x_{1}=0\})^{-1})}
  . 
  
    
      
        ◻
      
    
    {\displaystyle \Box }