[<< wikibooks] Geometry/Differential Geometry/Introduction
Differential geometry studies geometry by considering differentiable paramaterizations of curves, surfaces, and higher dimensional objects. Prerequisites include vector calculus, linear algebra, analysis, and topology.
One goal of differential geometry is to classify and represent differentiable curves in ways which are independent of their paramaterization. For example consider the curve represented by

y
=
3
x

{\displaystyle y=3x}
. Although

(
x
,
y
)
=
(
t
,
3
t
)

{\displaystyle (x,y)=(t,3t)}
and

(
x
,
y
)
=
(
3
t
,
9
t
)

{\displaystyle (x,y)=(3t,9t)}
are different paramterizations, they both represent the same curve. More generally, we consider the slope of the curve

3
=

d
y

d
x

=

d
y

d
t

⋅

1

d
x

d
t

=

3
1

{\displaystyle 3={\frac {dy}{dx}}={\frac {dy}{dt}}\cdot {\frac {1}{\frac {dx}{dt}}}={\frac {3}{1}}}
.
We call this type of curve a line. We can even rotate, and move it around, but it is still a line. The goal of Differential Geometry will be to similarly classify, and understand classes of differentiable curves, which may have different paramaterizations, but are still the same curve.
By adding sufficient dimensions, any equation can become a curve in geometry. Therefore, the ability to discern when two curves are unique also has the potential for applications in distinguishing information from noise. There may be multiple ways of receiving the same information--in different paramterizations, but we want to distinguish if the information is actually unique.