01 Key Concepts
Definition (2D)
For F(x,y) = (P,Q), divergence is div F = (partial P / partial x) + (partial Q / partial y).
Definition (3D)
For F(x,y,z) = (P,Q,R), div F = (partial P/partial x) + (partial Q/partial y) + (partial R/partial z).
Interpreting 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).
Zero Divergence
A field with divergence equal to zero everywhere is called incompressible — commonly used to model fluids that don't compress, like water.
Divergence Produces a Scalar
Unlike the gradient (which produces a vector), divergence always produces a single scalar value at each point.
02 Key Formulas
- div F = (partial P/partial x) + (partial Q/partial y) for F=(P,Q)
03 Solved Examples
Example 1 Find the divergence of F(x,y) = (x^2, y^2).
- P = x^2, so partial P/partial x = 2x.
- Q = y^2, so partial Q/partial y = 2y.
- Add them together.
Answer: div F = 2x + 2y
Example 2 Find the divergence of F(x,y) = (3x, -3y) and interpret its sign.
- partial P/partial x = 3. partial Q/partial y = -3.
- Sum: 3 + (-3) = 0.
Answer: div F = 0, so the field is incompressible everywhere
Example 3 Find the divergence of F(x,y,z) = (xy, yz, xz).
- partial P/partial x = y (from xy). partial Q/partial y = z (from yz). partial R/partial z = x (from xz).
- Add all three.
Answer: div F = y + z + x
04 Practice Questions
1Find the divergence of F(x,y) = (x, y).
2
2Find the divergence of F(x,y) = (2x, 3y).
5
3Find the divergence of F(x,y) = (y, x).
0
4Does zero divergence mean a field is a source or incompressible?
Incompressible
5Does divergence output a scalar or a vector?
A scalar
📄 Divergence — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.