Definition · applies to regions of space and manifolds

High school

Consider a region of the plane or of space. A vector field on the region is a rule that attaches a vector (an arrow) to each point of the region, such as the velocity of the wind at each point of a map.

Undergraduate

Consider an open set $U$ in $\reals^n$. A vector field on $U$ is a function $F : U \to \reals^n$, whose value $F(x)$ is thought of as a vector attached to the point $x$.

Graduate (differential geometry)

Consider a smooth manifold $M$ with tangent bundle $\pi : T M \to M$. A vector field on $M$ is a continuous map $X : M \to T M$, written $p \mapsto X_p$, that assigns to each point $p$ a vector in the tangent space at $p$:

\[X_p \in T_p M \quad \text{for all } p \in M.\]

Equivalently, $X$ is a section of $\pi$, that is, $\pi \circ X = \operatorname{id}_M$.

Properties

  • A vector field $f$ on $\reals^n$ defines the differential equation $\dot{x} = f(x)$, whose solutions are curves whose velocity at each point $x$ is $f(x)$. The flow of the vector field moves each point along these solutions.
  • A smooth vector field $X$ acts on smooth functions $h$ by differentiating them in the direction of $X$, $h \mapsto X h$, and this map is a derivation. The number $X h$ at a point is the Lie derivative of $h$ along $X$.
  • The Lie bracket $[X, Y]$ of two smooth vector fields $X$ and $Y$ is again a vector field.
  • The gradient of a differentiable function $h : \reals^n \to \reals$ is a vector field. In $\reals^3$, the gradient of every twice continuously differentiable function has zero curl, so a vector field with nonzero curl, such as $(x, y, z) \mapsto (-y, x, 0)$, is not a gradient.

Notes

A vector field is different from a field in algebra, such as the real numbers, which is a set of numbers with addition and multiplication.

It is also different from a scalar field, in physics, which assigns a number to each point, such as the temperature at each point of a room.

In differential geometry, the coordinates of a vector field depend on the coordinate system: in a chart $(x^1, \dots, x^n)$, a vector field is written $X = X^i \frac{\partial}{\partial x^i}$, where the functions $X^i$ are its component functions.

A field of covectors is a differential form of degree $1$, not a vector field.

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