Vector fields

Tangent vectors

Definition 1. A tangent vector $V$ at $p \in S$ (a differentiable manifold) is an equivalence class in the set of curves $γ : (- ϵ, ϵ) \to S$ such that $γ (0) = p$ . To compare two curves we fix a chart $(U, φ)$ around $p$ and use it to differentiate: $γ_{1}$ and $γ_{2}$ are equivalent if they have the same velocity at $t = 0$ , i.e., if

(φ \circ γ_{1})^{'} (0) = (φ \circ γ_{2})^{'} (0) .

It can be proven that this does not depend on the chart (changing chart applies the same invertible linear map to both velocities).
Also, $V$ can be defined as an operator from $C^{\infty} (S)$ to $R$

V (f) = (f \circ γ)^{'} (0)

Definition 2. The tangent vector space $T_{p} S$ is the set of all tangent vectors at $p$ . It is a vector space, with the addition and scalar multiplication of tangent vectors defined pointwise, and zero vector given by the equivalence class of the constant curve at $p$ .

Definition 3. The differential of a map $F : S \to R^{m}$ at a point $p \in S$ is a linear map $d F_{p} : T_{p} S \to R^{m}$ defined as follows: for any tangent vector $V \in T_{p} S$ , we define $d F_{p} (V)$ as a vector whose action on a function $g$ in $C^{\infty} (R^{m}$ is

d F_{p} (V) (g) = V (g \circ F)

(observe that $g \circ F \in C^{\infty} (S)$ ).

With all this three ingredients, you have that $d F_{p}$ is linear since

d F_{p} (V + W) (g) = (V + W) (g \circ F) = V (g \circ F) + W (g \circ F) =

= d F_{p} (V) (g) + d F_{p} (W) (g)

and

d F_{p} (λ V) (g) = λ V (g \circ F) = λ d F_{p} (V) (g)

Why do we define tangent vectors as derivations:

Motivation. In $R^{n}$ an ordinary vector $v$ at $p$ is the same as the directional derivative $D_{v} f = \frac{d}{d t} |_{0} f (p + t v)$ . So one can think of a vector as a derivation: an operator on functions that is linear and satisfies Leibniz.
Definition. We define tangent vectors at $p$ in a manifold $M$ as derivations of (the germs of) smooth functions at $p$ , i.e., operators $D$ with

D (f g) = f (p) D (g) + g (p) D (f) .

The collection of all of them constitutes the tangent space at $p$ : $T_{p} M$ .
From curves to vectors. Given a smooth curve $γ$ such that $γ (0) = p$ , the formula

γ^{'} (0) (f) = \frac{d}{d t} (f \circ γ) (0)

defines a derivation, so every curve through $p$ gives a tangent vector.

A basis. If we take as particular $γ$ the coordinate curves of a chart $φ \equiv (x_{1}, x_{2}, \dots, x_{n})$ we obtain $n$ vectors that constitute a basis of $T_{p} M$ , denoted by $\partial_{x_{1}}, \partial_{x_{2}}, \dots, \partial_{x_{n}}$ . This is where the chart is really used, and it shows $\dim T_{p} M = n$ .
Converse. Every derivation $D = \sum_{i} a_{i} \partial_{x_{i}} |_{p}$ is the velocity of the curve

t \mapsto φ^{- 1} (φ (p) + t (a_{1}, \dots, a_{n}))

(not unique). This gives the second characterization: tangent vectors are equivalence classes of curves through $p$ with the same velocity.

There exists a third characterization (more abstract): by assigning to each chart around $p$ a vector of $R^{n}$ , subject to a contravariant change rule between charts. Related to associated bundles, I think.
Fourth: if $M$ sits inside $R^{N}$ , which is always possible by Whitney embedding theorem, then tangent vectors can be seen as the usual vectors of $R^{N}$ .

Vector fields

Smooth sections of the tangent bundle: $T S = \cup_{p \in S} {p} \times T_{p} S$ .

Important properties:
the canonical form of a regular vector field,
the canonical form of commuting vector fields and
the flow of a vector field.