Divergence and Curl

Two local measurements

Divergence measures net outward flow per unit volume near a point. Curl measures local circulation per unit area, with its direction set by the right-hand rule. Neither is the magnitude of the field itself. A large uniform velocity has zero divergence and zero curl.

Visual guide

VISUAL GUIDEExpansion and rotation are different field patterns

Source: divergence

Rotation: curl

The left field (x, y) flows outward and has planar divergence 2 with zero curl. The right field (−y, x) circulates and has curl component 2 with zero divergence. Arrows are normalized to show direction, not speed.

Compute from components

For $\mathbf F=(P,Q,R)$ ,

\nabla\cdot\mathbf F=P_x+Q_y+R_z.

\nabla\times\mathbf F=(R_y-Q_z,P_z-R_x,Q_x-P_y).

Divergence is a scalar; curl is a vector. In a planar field $(P,Q,0)$ independent of $z$ , only the curl’s $z$ component survives. When second partial derivatives are continuous, $\nabla\times\nabla f=0$ and $\nabla\cdot(\nabla\times\mathbf F)=0$ .

Worked example: a source

For $\mathbf F=(x,y,z)$ , divergence is $3$ and curl is zero. Across a small cube of side $h$ , opposite faces contribute a net $h^3$ in each coordinate direction, giving total outward flux $3h^3$ . Dividing by volume recovers $3$ .

Worked example: rotation without expansion

For $\mathbf F=(-y,x,0)$ , divergence is zero and curl is $(0,0,2)$ . The vectors are tangent to circles about the $z$ axis. A unit square in the $xy$ plane has counterclockwise circulation $2$ . The fluid rotates without net local volume creation.

Interpreting a zero

Zero divergence does not force a field to vanish: material can enter one side and leave another. Zero curl does not by itself guarantee a global potential on a domain with holes. These differential measurements become total flux and total circulation through the integral theorems later in the unit.

Practice

Find divergence and curl of $(2x,-y,3z)$ .
Repeat for the constant field $(4,1,-2)$ .
Can $(x,y,z)$ be the curl of a smooth field on an open region?

Show worked solutions

Divergence $2-1+3=4$ ; every cross derivative is zero, so curl is zero.
All derivatives vanish, giving both zero despite nonzero velocity.
No. Every smooth curl has zero divergence, whereas this field has divergence $3$ .

Further study

MIT OpenCourseWare: Multivariable Calculus provides a full university course with additional lectures and exercises.