Vector field

Let $M$ be a smooth manifold and let $\pi:TM\to M$ denote its tangent bundle .

Definition. A (smooth) vector field on $M$ is a smooth map $X:M\to TM$ such that $\pi\circ X=\mathrm{id}_M$. Equivalently, $X$ is a smooth section of the tangent bundle, assigning to each $p\in M$ a tangent vector

(where $T_pM$ is the tangent space at $p$ ) in a way that is smooth in local coordinates.

A vector field can also be viewed as a derivation on smooth functions: for each $X$ and each $f\in C^\infty(M)$, one obtains a smooth function $X(f)\in C^\infty(M)$ defined by differentiating $f$ in the direction $X$. Using the pairing between tangent and cotangent spaces (see the cotangent bundle ), this can be written pointwise as

Given a smooth map $F:M\to N$, the differential (pushforward) $dF_p$ transports tangent vectors, and when $F$ is a diffeomorphism one can push vector fields forward along $F$ by applying $dF$ pointwise.

Examples

1. Coordinate vector fields on $\mathbb{R}^n$. On $M=\mathbb{R}^n$ with coordinates $(x^1,\dots,x^n)$, the vector field $\partial/\partial x^i$ is defined by

   $$\left(\frac{\partial}{\partial x^i}\right)_p(f)=\frac{\partial (f\circ x^{-1})}{\partial x^i}\bigg|_{x(p)}.$$

   More generally, any $X=\sum_{i=1}^n a_i(x)\,\partial/\partial x^i$ with smooth coefficient functions $a_i$ is a smooth vector field.

2. Radial and rotational fields on $\mathbb{R}^2$. In coordinates $(x,y)$, the radial field

   $$X = x\,\frac{\partial}{\partial x}+y\,\frac{\partial}{\partial y}$$

   points outward from the origin, while the rotational field

   $$Y = -y\,\frac{\partial}{\partial x}+x\,\frac{\partial}{\partial y}$$

   generates counterclockwise rotation (away from the origin it is tangent to circles).

3. Left-invariant vector fields on a Lie group. If $G$ is a Lie group , each element of the Lie algebra determines a unique left-invariant vector field by translating that tangent vector from the identity to every point via left translation .