Divergence and curl

Net outflow per unit area

Our running example is a velocity field. It might be the flow of a fluid, or the joint-velocity field of the two-link arm: at each configuration, the arrows tell you how the hand is moving. Differentiating a scalar function gave one new function. Differentiating a field gives two scalar fields. The first measures how much the flow spreads apart at a point.

Take a tiny square $\square_h$ of half-width $h$ centred on a probe $p$. The flux of $F$ through its boundary is the total amount of field crossing outward, and the divergence is that flux per unit area in the limit of a vanishing box:

On each of the four sides the outward normal $n$ is constant, so the integral is just $F\cdot n$ times the side length. With $F=(P,Q)$ that gives the coordinate formula — the sum of the two partial derivatives that measure how each component grows along its own direction:

Pick a preset and drag the probe. The readout computes the flux over the four sides, divides by the box area $4h^2$, and prints it next to the analytic value. The arrows at the boundary are the outward-normal components: they point out where the field leaves, in where it enters.

Net circulation per unit area

The second derivative asks a different question: if a tiny pinwheel sat at $p$, would the flow turn it? Take a small disc $D_h$ of radius $h$, and add up the component of the field along the tangent to its rim. That is the circulation, and the curl is circulation per unit area as the disc shrinks:

Sample the rim and add the tangential components and the same two-line formula falls out, now with the opposite pairing of partials:

Drag the probe and watch the tangential arrows on the rim. On rotation $F=(-y,x)$ every arrow pushes the wheel the same way and the curl is a steady $+2$. On shear $F=(y,0)$ the top and bottom of the wheel are pushed in opposite directions, so the net circulation is negative. The readout compares the sampled circulation over the rim with the analytic value.

Divergence and curl as heatmaps

Because both are scalars, we can shade the plane by their value. The top map is $\operatorname{div} F$, the bottom is $\operatorname{curl} F$ — magenta for positive, teal for negative, near-white near zero. Switching presets and dragging the probe shows that the field and its two derivative fields are different pictures of the same object.

A source is positive divergence everywhere and zero curl; a rotation is the mirror image, zero divergence and positive curl; a saddle and the uniform drift are zero in both. The readout tracks both numbers at the probe as it moves across the shaded plane.

The limit is what makes them derivatives

Divergence and curl were defined as limits, and the whole point of the limit is that the box and the wheel forget their own size. Slide $h$ down and the numeric estimate walks onto the analytic value: the probe stops measuring the field around the point and starts measuring the field at the point.

The readout shows both the signed error and the error divided by $h^2$. For the smooth nonlinear field below that ratio settles onto a constant, which is exactly what an $h^2$ error looks like; a cruder one-sided difference would instead have error $O(h)$, with the error divided by $h$ settling. Either way the estimate converges, and that is the content of “these are derivatives.”

Incompressible and irrotational, and what each buys you

The two numbers name two special kinds of field. If $\operatorname{div}F=0$ everywhere the flow is incompressible: no point creates or destroys fluid, so a small blob keeps its area as it is carried along. If $\operatorname{curl}F=0$ everywhere the field is irrotational, and on a simply connected domain that is exactly the condition for it to be conservative — there is a potential $\phi$ with $F=\nabla\phi$. That is the test Part 2 built up by dragging paths, now reduced to one cheap partial-derivative check.

Toggle between a purely rotational field (divergence-free), a purely radial one (curl-free), and one with both nonzero. The readout says which of the two numbers vanishes, and a field that is both is potential flow: $F=\nabla\phi$ with $\nabla^2\phi=0$.

The densities the integral theorems integrate

Divergence and curl are not ends in themselves — they are the densities that the big integral theorems integrate. Green's, Stokes' and the divergence theorem all say the same thing: the behaviour on a boundary equals the accumulated behaviour inside, and the thing being accumulated is exactly one of these two scalars. That is Part 4: Green, Stokes, Divergence: one theorem, and this page is its integrand. The swirl you measured with the paddlewheel is what Stokes integrates over a surface; the outflow you measured with the box is what the divergence theorem integrates over a volume.

Two more places the picture returns. A dynamical system $\dot p=F(p)$ is a velocity field, and Part 12: Differential equations uses the divergence to decide whether a flow stretches or squeezes phase-space volume. And the gradient fields that fill Nonlinear Optimization are curl-free by construction, $\nabla\times\nabla f=0$: a loss surface has no rotational swirl, which is why descent paths cannot loop back on themselves in the way a general flow can.

• 3Blue1Brown, Divergence and curl — the same box and paddlewheel, animated.
• MIT 18.02SC, Multivariable Calculus — the course that runs alongside this volume, fields and all.
• Khan Academy, Multivariable calculus — extra practice on divergence, curl and the theorems that follow.