Vector Fields

A vector field on a region assigns a vector $\mathbf{F}(x, y) = \langle P(x, y), Q(x, y)\rangle$ (or $\langle P, Q, R\rangle$ in 3D) to every point. You've already met a few in 14.09.

Examples

Field	Picture
$\langle x, y\rangle$	arrows pointing straight out from the origin, longer farther away
$\langle -y, x\rangle$	arrows circling counterclockwise
$-\dfrac{\mathbf{r}}{\lvert\mathbf{r}\rvert^3}$	gravity: arrows pointing toward the origin, stronger up close
$\langle 1, 0\rangle$	a steady wind blowing east

Gradient fields

For any function $f$, $\nabla f$ is a vector field (14.08), pointing uphill. A field that's a gradient, $\mathbf{F} = \nabla f$, is called conservative, and $f$ is its potential function. For example, $\langle 2x, 2y\rangle = \nabla(x^2 + y^2)$. Gravity is conservative: it's $\nabla\frac{1}{\lvert\mathbf{r}\rvert}$ (up to constants), and physicists call $-f$ the potential energy.

Not every field is a gradient: the swirling $\langle -y, x\rangle$ isn't (its curl is $2$, not $0$). Sections 16.04–16.05 explain why it matters.

Flow lines

A flow line (or streamline) is a path that follows the arrows: its velocity at each point is $\mathbf{F}$ there. A leaf dropped in a river traces one. For $\langle -y, x\rangle$, the flow lines are circles around the origin; for $\langle 1, 0\rangle$, horizontal lines.