Vector Fields

Vector fields assign a vector to each point in space, modeling physical phenomena such as fluid flow, gravitational fields, and electromagnetic fields.

Definition and Examples

Definition

A vector field on a region $D \subseteq \mathbb{R}^n$ is a function $\mathbf{F} : D \to \mathbb{R}^n$ that assigns to each point $\mathbf{x} \in D$ a vector $\mathbf{F}(\mathbf{x})$ . In $\mathbb{R}^2$ , $\mathbf{F}(x,y) = P(x,y)\,\mathbf{i} + Q(x,y)\,\mathbf{j}$ . In $\mathbb{R}^3$ , $\mathbf{F}(x,y,z) = P\,\mathbf{i} + Q\,\mathbf{j} + R\,\mathbf{k}$ .

ExamplePhysical vector fields

Gravitational field: $\mathbf{F} = -\frac{GM}{r^3}\mathbf{r}$ where $\mathbf{r} = (x,y,z)$ and $r = \|\mathbf{r}\|$
Velocity field of a rotating body: $\mathbf{F} = \boldsymbol{\omega} \times \mathbf{r} = (-\omega y, \omega x, 0)$
Electric field of a point charge: $\mathbf{E} = \frac{q}{4\pi\epsilon_0 r^3}\mathbf{r}$

Gradient, Divergence, and Curl

Definition

The three fundamental differential operators of vector calculus are:

Gradient: $\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)$ (scalar $\to$ vector)
Divergence: $\nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}$ (vector $\to$ scalar)
Curl: $\nabla \times \mathbf{F} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right)\mathbf{i} + \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\right)\mathbf{j} + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\mathbf{k}$ (vector $\to$ vector)

Conservative Fields

Definition

A vector field $\mathbf{F}$ is conservative if $\mathbf{F} = \nabla f$ for some scalar function $f$ , called the potential function. Equivalently (on simply connected domains), $\mathbf{F}$ is conservative if and only if $\nabla \times \mathbf{F} = \mathbf{0}$ .

RemarkTwo key identities

The fundamental identities $\nabla \times (\nabla f) = \mathbf{0}$ (curl of gradient is zero) and $\nabla \cdot (\nabla \times \mathbf{F}) = 0$ (divergence of curl is zero) reflect the deep algebraic structure of the exterior derivative in differential forms, where they correspond to $d^2 = 0$ .