Calculus · Quick Reference

Divergence Cheat Sheet

Multivariable Calculus · Lesson 19/20

In one line: divergence measures how much a vector field is spreading out from (or converging into) a given point — useful for describing sources and sinks in flow fields.

Key Ideas

1Definition (2D). For F(x,y) = (P,Q), divergence is div F = (partial P / partial x) + (partial Q / partial y).

2Definition (3D). For F(x,y,z) = (P,Q,R), div F = (partial P/partial x) + (partial Q/partial y) + (partial R/partial z).

3Interpreting the Sign. Positive divergence at a point suggests the field is 'expanding' outward there (a source); negative divergence suggests it's 'converging' inward (a sink).

4Zero Divergence. A field with divergence equal to zero everywhere is called incompressible — commonly used to model fluids that don't compress, like water.

5Divergence Produces a Scalar. Unlike the gradient (which produces a vector), divergence always produces a single scalar value at each point.

Worked Examples

Find the divergence of F(x,y) = (x^2, y^2).

div F = 2x + 2y

Find the divergence of F(x,y) = (3x, -3y) and interpret its sign.

div F = 0, so the field is incompressible everywhere

Find the divergence of F(x,y,z) = (xy, yz, xz).

div F = y + z + x

Formulas

div F = (partial P/partial x) + (partial Q/partial y) for F=(P,Q)

Practice Yourself

Find the divergence of F(x,y) = (x, y).

Find the divergence of F(x,y) = (2x, 3y).

Find the divergence of F(x,y) = (y, x).