Editing Tangent space (section)

== Properties ==

If <math> M </math> is an open subset of <math> \mathbb{R}^{n} </math>, then <math> M </math> is a <math> C^{\infty} </math> manifold in a natural manner (take coordinate charts to be [[Identity function|identity maps]] on open subsets of <math> \mathbb{R}^{n} </math>), and the tangent spaces are all naturally identified with <math> \mathbb{R}^{n} </math>.

=== Tangent vectors as directional derivatives ===

Another way to think about tangent vectors is as [[directional derivative]]s. Given a vector <math> v </math> in <math> \mathbb{R}^{n} </math>, one defines the corresponding directional derivative at a point <math> x \in \mathbb{R}^{n} </math> by
:<math>
\forall f \in {C^{\infty}}(\mathbb{R}^{n}): \qquad
  (D_{v} f)(x) :=
  \left. \frac{\mathrm{d}}{\mathrm{d}{t}} [f(x + t v)] \right|_{t = 0}
= \sum_{i = 1}^{n} v^{i} {\frac{\partial f}{\partial x^{i}}}(x).
</math>
This map is naturally a derivation at <math> x </math>. Furthermore, every derivation at a point in <math> \mathbb{R}^{n} </math> is of this form. Hence, there is a one-to-one correspondence between vectors (thought of as tangent vectors at a point) and derivations at a point.

As tangent vectors to a general manifold at a point can be defined as derivations at that point, it is natural to think of them as directional derivatives. Specifically, if <math> v </math> is a tangent vector to <math> M </math> at a point <math> x </math> (thought of as a derivation), then define the directional derivative <math> D_{v} </math> in the direction <math> v </math> by
:<math>
\forall f \in {C^{\infty}}(M): \qquad
{D_{v}}(f) := v(f).
</math>
If we think of <math> v </math> as the initial velocity of a differentiable curve <math> \gamma </math> initialized at <math> x </math>, i.e., <math> v = \gamma'(0) </math>, then instead, define <math> D_{v} </math> by
:<math>
\forall f \in {C^{\infty}}(M): \qquad
{D_{v}}(f) := (f \circ \gamma)'(0).
</math>

=== Basis of the tangent space at a point ===

For a <math> C^{\infty} </math> manifold <math> M </math>, if a chart <math> \varphi = (x^{1},\ldots,x^{n}): U \to \mathbb{R}^{n} </math> is given with <math> p \in U </math>, then one can define an ordered basis <math display="inline"> \left\{ \left. \frac{\partial}{\partial x^{1}} \right|_{p} , \dots , \left. \frac{\partial}{\partial x^{n}} \right|_{p} \right\} </math> of <math> T_{p} M </math> by
:<math>
\forall i \in \{ 1,\ldots,n \}, ~ \forall f \in {C^{\infty}}(M): \qquad
{ \left. \frac{\partial}{\partial x^{i}} \right|_{p}}(f) :=
\left( \frac{\partial}{\partial x^{i}} \Big( f \circ \varphi^{- 1} \Big) \right) \Big( \varphi(p) \Big) .
</math>
Then for every tangent vector <math> v \in T_{p} M </math>, one has
:<math>
v = \sum_{i = 1}^{n} v^{i} \left. \frac{\partial}{\partial x^{i}} \right|_{p}.
</math>
This formula therefore expresses <math> v </math> as a linear combination of the basis tangent vectors <math display="inline"> \left. \frac{\partial}{\partial x^{i}} \right|_{p} \in T_{p} M </math> defined by the coordinate chart <math> \varphi: U \to \mathbb{R}^{n} </math>.<ref>{{cite web|title = An Introduction to Differential Geometry|first = Eugene|last = Lerman|page = 12| url = https://faculty.math.illinois.edu/~lerman/518/f11/8-19-11.pdf}}</ref>

=== The derivative of a map ===

{{main|Pushforward (differential)}}

Every smooth (or differentiable) map <math> \varphi: M \to N </math> between smooth (or differentiable) manifolds induces natural [[linear map]]s between their corresponding tangent spaces:
:<math>
\mathrm{d}{\varphi}_{x}: T_{x} M \to T_{\varphi(x)} N.
</math>
If the tangent space is defined via differentiable curves, then this map is defined by
:<math>
{\mathrm{d}{\varphi}_{x}}(\gamma'(0)) := (\varphi \circ \gamma)'(0).
</math>
If, instead, the tangent space is defined via derivations, then this map is defined by
:<math>
[\mathrm{d}{\varphi}_{x}(D)](f) := D(f \circ \varphi).
</math>

The linear map <math> \mathrm{d}{\varphi}_{x} </math> is called variously the ''derivative'', ''total derivative'', ''differential'', or ''pushforward'' of <math> \varphi </math> at <math> x </math>. It is frequently expressed using a variety of other notations:
:<math>
D \varphi_{x}, \qquad (\varphi_{*})_{x}, \qquad \varphi'(x).
</math>
In a sense, the derivative is the best linear approximation to <math> \varphi </math> near <math> x </math>. Note that when <math> N = \mathbb{R} </math>, then the map <math> \mathrm{d}{\varphi}_{x}: T_{x} M \to \mathbb{R} </math> coincides with the usual notion of the [[Differential (calculus)|differential]] of the function <math> \varphi </math>. In [[local coordinates]] the derivative of <math> \varphi </math> is given by the [[Jacobian matrix and determinant|Jacobian]].

An important result regarding the derivative map is the following:
{{math theorem|math_statement=If <math> \varphi: M \to N </math> is a [[local diffeomorphism]] at <math> x </math> in <math> M </math>, then <math> \mathrm{d}{\varphi}_{x}: T_{x} M \to T_{\varphi(x)} N </math> is a linear [[isomorphism]]. Conversely, if <math>\varphi : M\to N</math> is continuously differentiable and <math>\mathrm{d}{\varphi}_{x}</math> is an isomorphism, then there is an [[open set|open neighborhood]] <math> U </math> of <math> x </math> such that <math> \varphi </math> maps <math> U </math> diffeomorphically onto its image.}}
This is a generalization of the [[inverse function theorem]] to maps between manifolds.