Vector field

A vector field assigns a vector to each point in space. In 3D:

$\mathbf{F}:\Omega \subseteq {\mathbb{R}}^3\to {\mathbb{R}}^3,\mathbf{F}(x,y,z)=P(x,y,z)\hat{\mathbf{i}}+Q(x,y,z)\hat{\mathbf{j}}+R(x,y,z)\hat{\mathbf{k}}.$

In 2D: $\mathbf{F}(x,y)=⟨P(x,y),Q(x,y)⟩$.

A vector field is not one specific vector. It’s an infinite collection, one at every point. Any rule that assigns vectors to points is a valid vector field.

Distinct from vector-valued functions of one variable

A vector-valued function $\mathbf{r}(t)$ assigns a vector to each parameter value, so the vectors live along a curve. A vector field assigns a vector to each point in space. Same kind of output (vectors), different objects: $\mathbf{r}$ traces out a curve, $\mathbf{F}$ fills space.

Examples everywhere in physics

• Gravitational field: gravitational force per unit mass at each point.
• Electric field: force per unit positive test charge at each point.
• Magnetic field: a vector at each point in space.
• Velocity field of a fluid: at each point, the velocity of the fluid that’s there.
• Gradient of a scalar function (temperature, pressure): direction of steepest increase. See Gradient and Gradient field.

Simple 2D fields to recognize

• Constant field $\mathbf{F}=⟨1,0⟩$: uniform grid of identical arrows.
• Radial field $\mathbf{F}=⟨x,y⟩$: arrows point outward from origin, magnitude $\sqrt{x^2+y^2}$.
• Rotational field $\mathbf{F}=⟨-y,x⟩$: at $(1,0)$ the vector is $⟨0,1⟩$; at $(0,1)$ it’s $⟨-1,0⟩$. Arrows rotate counterclockwise around the origin.
• Sink $\mathbf{F}=⟨-x,-y⟩$: arrows point radially inward.
• Inverse-square $\mathbf{F}=\mathbf{r}/|\mathbf{r}{|}^3$: points radially outward, magnitude $1/|\mathbf{r}{|}^2$. See Inverse-square field.

Most fields you’ll see are combinations of these.

Visualization

For 2D fields, draw the vector $\mathbf{F}(x,y)$ as an arrow with tail at $(x,y)$, for a grid of sample points. 3D is harder: use 2D slices, streamlines, or color maps of magnitude. Stream plots (curves tangent to the field) usually read better than arrow plots for messier fields.

Operations on vector fields

A vector field is the input for:

• The Divergence $\nabla \cdot \mathbf{F}$ (scalar field, “source density”).
• The Curl $\nabla \times \mathbf{F}$ (vector field, “circulation density”).
• Line integrals ${\int }_C\mathbf{F}\cdot d\mathbf{r}$ along curves (work / circulation).
• Flux integrals ${\iint }_S\mathbf{F}\cdot d\mathbf{S}$ across surfaces.

These four operations, plus the theorems relating them (Green’s theorem, Stokes’ theorem, Divergence theorem), are most of classical vector calculus.

Important special classes

• Conservative (or gradient) fields: $\mathbf{F}=\nabla \varphi$ for some scalar $\varphi$. Equivalent to zero curl on simply connected domains. See Conservative vector field.
• Source-free (or solenoidal) fields: $\nabla \cdot \mathbf{F}=0$. Equivalent to having a stream function (in 2D) or vector potential (in 3D).
• Harmonic / irrotational and source-free: both at once. In 2D, these show up as the real and imaginary parts of analytic functions, which is where vector calculus meets complex analysis.