Divergence and Curl

A vector field \mathbf{F}(x,y,z) = (P, Q, R) attaches an arrow to every point of space — a velocity of flowing water, an electric field, a wind map. The operator \nabla (called "del" or "nabla")

\nabla = \left(\frac{\partial}{\partial x},\ \frac{\partial}{\partial y},\ \frac{\partial}{\partial z}\right)

is a vector of partial derivatives. Combine it with a field in the two natural ways a vector combines with another vector — the dot product and the cross product — and you get the two first-order ways a field can vary: how much it spreads, and how much it spins.

Divergence: net outflow per unit volume

The divergence is the dot product \nabla \cdot \mathbf{F} — a scalar at each point measuring the net rate at which the field flows out of a tiny region around it. Treating \nabla as a vector and dotting it with (P, Q, R):

\operatorname{div}\mathbf{F} = \nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}.

Positive divergence at a point marks a source (more flows out than in); negative marks a sink; zero means whatever flows in flows back out (an incompressible flow).

Worked example — divergence

Take the outward "explosion" field \mathbf{F} = (x,\ y,\ z), every arrow pointing radially away from the origin.

Step 1 — read off the components. P = x, Q = y, R = z.

Step 2 — differentiate each by its matching variable.

\frac{\partial P}{\partial x} = \frac{\partial x}{\partial x} = 1, \qquad \frac{\partial Q}{\partial y} = 1, \qquad \frac{\partial R}{\partial z} = 1.

Step 3 — add.

\nabla \cdot \mathbf{F} = 1 + 1 + 1 = 3.

A constant +3 everywhere: this field is a source at every point, spreading uniformly. (In two dimensions the same (x,y) would give 2 — one per spatial dimension.)

Curl: local rotation

The curl is the cross product \nabla \times \mathbf{F} — a vector at each point whose direction is the axis of local spin (by the right-hand rule) and whose length is twice the local angular speed. Expanding the formal determinant,

\operatorname{curl}\mathbf{F} = \nabla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\[2pt] \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\[4pt] P & Q & R \end{vmatrix} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z},\ \ \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x},\ \ \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right).

If you placed a tiny paddle wheel in the flow, the curl tells you which way and how fast it would turn. A field with zero curl everywhere is called irrotational.

Worked example — curl

Take the "merry-go-round" field \mathbf{F} = (-y,\ x,\ 0), which circulates counter-clockwise about the z-axis.

Step 1 — components. P = -y, Q = x, R = 0.

Step 2 — the \mathbf{i} component, \partial R/\partial y - \partial Q/\partial z. Both R = 0 and Q = x are free of z and y here:

\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} = 0 - 0 = 0.

Step 3 — the \mathbf{j} component, \partial P/\partial z - \partial R/\partial x:

\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} = 0 - 0 = 0.

Step 4 — the \mathbf{k} component, \partial Q/\partial x - \partial P/\partial y:

\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = \frac{\partial x}{\partial x} - \frac{\partial(-y)}{\partial y} = 1 - (-1) = 2.

Step 5 — assemble.

\nabla \times \mathbf{F} = (0,\ 0,\ 2) = 2\,\mathbf{k}.

The curl points along +\mathbf{k} — the spin axis of a counter-clockwise swirl in the xy-plane — and its magnitude 2 is twice the angular speed 1. Notice this field has \nabla \cdot \mathbf{F} = -0 + 0 + 0 = 0: pure rotation, no spreading.

Two identities: "div of curl" and "curl of grad"

Mixed second derivatives commute (Clairaut's theorem), \partial^2/\partial x\,\partial y = \partial^2/\partial y\,\partial x. That single fact forces two clean cancellations.

(a) The divergence of a curl is zero

Step 1 — take the divergence of the curl. Dot \nabla into the curl vector from above:

\nabla \cdot (\nabla \times \mathbf{F}) = \frac{\partial}{\partial x}\!\left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right) + \frac{\partial}{\partial y}\!\left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\right) + \frac{\partial}{\partial z}\!\left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right).

Step 2 — expand into six terms.

= R_{yx} - Q_{zx} + P_{zy} - R_{xy} + Q_{xz} - P_{yz}.

Step 3 — pair them off. By Clairaut, R_{yx} = R_{xy}, Q_{zx} = Q_{xz}, and P_{zy} = P_{yz}, so the six terms cancel in three pairs:

\nabla \cdot (\nabla \times \mathbf{F}) = 0.

Curls never have sources — a swirl neither creates nor destroys flow.

(b) The curl of a gradient is zero

Step 1 — start from a scalar field's gradient \nabla f = (f_x,\ f_y,\ f_z), and take its curl. The \mathbf{k} component is

(\nabla \times \nabla f)_{\mathbf{k}} = \frac{\partial}{\partial x}(f_y) - \frac{\partial}{\partial y}(f_x) = f_{yx} - f_{xy}.

Step 2 — apply Clairaut. f_{yx} = f_{xy}, so this component is 0; the same cancellation kills the \mathbf{i} and \mathbf{j} components.

\nabla \times (\nabla f) = \mathbf{0}.

Gradient fields never swirl — they are exactly the irrotational (conservative) fields. These two identities are the algebraic skeleton behind Stokes' theorem and the divergence theorem.

For a vector field \mathbf{F} = (P, Q, R) with continuous second partials and a scalar field f:

Two imaginary probes make the operators physical. Drop a tiny paddle wheel into the flow: it spins about the curl vector at half its length, so it stays still exactly where the field is irrotational and whirls fastest where the curl is largest. The (-y, x, 0) merry-go-round above spins it about +\mathbf{k}; the radial (x,y,z) leaves it dead still.

Now imagine a tiny balloon drifting with the flow. Where the divergence is positive the surrounding field carries its skin outward and the balloon inflates; where it is negative the balloon shrinks; where it is zero the volume is preserved even as the shape distorts. Divergence measures change of size, curl measures change of orientation — together they capture the entire first-order behaviour of a flow.

See it: spread vs spin

Switch between the two archetypes. The radial field \mathbf{F} = (x, y) is all divergence and no curl; the circulating field \mathbf{F} = (-y, x) is all curl and no divergence. The computed values appear beneath each.