01 Key Concepts
Definition (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).
Definition (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.
Interpreting Curl
A nonzero curl at a point indicates the field is rotating there, like a tiny paddlewheel would spin if placed in the flow.
Zero Curl
A field with curl equal to zero everywhere is called irrotational — it has no local spinning tendency anywhere.
Relationship to Divergence
Curl and divergence both measure different behaviors of a vector field: divergence measures spreading/shrinking, curl measures rotation.
02 Key Formulas
- 2D curl F = (partial Q/partial x) - (partial P/partial y) for F=(P,Q)
03 Solved Examples
Example 1 Find the 2D curl of F(x,y) = (-y, x).
- P = -y, so partial P/partial y = -1.
- Q = x, so partial Q/partial x = 1.
- Curl = partial Q/partial x - partial P/partial y = 1 - (-1).
Answer: curl F = 2 (a strong rotational field)
Example 2 Find the 2D curl of F(x,y) = (x, y) and interpret it.
- P = x, so partial P/partial y = 0.
- Q = y, so partial Q/partial x = 0.
- Curl = 0 - 0.
Answer: curl F = 0, so the field is irrotational (no swirling)
Example 3 Find the 2D curl of F(x,y) = (y^2, x^2).
- P = y^2, so partial P/partial y = 2y.
- Q = x^2, so partial Q/partial x = 2x.
- Curl = 2x - 2y.
Answer: curl F = 2x - 2y
04 Practice Questions
1Find the 2D curl of F(x,y) = (y, -x).
-2
2Find the 2D curl of F(x,y) = (2x, 2y).
0
3Find the 2D curl of F(x,y) = (x^2, y).
0
4Does zero curl mean a field is rotational or irrotational?
Irrotational
5Does divergence measure spreading or rotation?
Spreading (curl measures rotation)
📄 Curl — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.