Calculus · Quick Reference

Curl Cheat Sheet

Multivariable Calculus · Lesson 20/20

In one line: curl measures the rotation or 'swirling' tendency of a vector field at a point — how much the field circulates around that point.

Key Ideas

1Definition (2D scalar curl). For F(x,y) = (P,Q), the 2D curl (a scalar) is curl F = (partial Q/partial x) - (partial P/partial y).

2Definition (3D vector curl). For F(x,y,z)=(P,Q,R), curl F is a vector formed from differences of partial derivatives across all three pairs of components.

3Interpreting Curl. A nonzero curl at a point indicates the field is rotating there, like a tiny paddlewheel would spin if placed in the flow.

4Zero Curl. A field with curl equal to zero everywhere is called irrotational — it has no local spinning tendency anywhere.

5Relationship to Divergence. Curl and divergence both measure different behaviors of a vector field: divergence measures spreading/shrinking, curl measures rotation.

Worked Examples

Find the 2D curl of F(x,y) = (-y, x).

curl F = 2 (a strong rotational field)

Find the 2D curl of F(x,y) = (x, y) and interpret it.

curl F = 0, so the field is irrotational (no swirling)

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

curl F = 2x - 2y

Formulas

2D curl F = (partial Q/partial x) - (partial P/partial y) for F=(P,Q)

Practice Yourself

Find the 2D curl of F(x,y) = (y, -x).

-2

Find the 2D curl of F(x,y) = (2x, 2y).

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