Editing Tangent space (section)

=== 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>