Curl Calculator

Curl measures how much a vector field rotates around a point. The direction of the curl vector is the axis of that rotation (following the right-hand rule), and its magnitude gives the angular speed of a tiny paddle wheel placed in the flow.

For F = (P, Q, R), the curl is:

curl F = (dR/dy - dQ/dz, dP/dz - dR/dx, dQ/dx - dP/dy)

This calculator uses a linear vector field where each component depends only on cross-variables (not its own):

P(y,z) = ay + bz Q(x,z) = cx + dz R(x,y) = ex + fy

The partial derivatives are all constants: dP/dy = a, dP/dz = b, dQ/dx = c, dQ/dz = d, dR/dx = e, dR/dy = f. Plugging in:

curl F = (f - d, b - e, c - a)

For a linear field like this, the curl is the same everywhere — it does not depend on the evaluation point. That is a special property of linear fields.

Worked example: P = y + 2z, Q = 3x + z, R = x + 4y. So a=1, b=2, c=3, d=1, e=1, f=4. curl F = (4-1, 2-1, 3-1) = (3, 1, 2).

The curl vector (3, 1, 2) points in the direction of the rotation axis. Its magnitude sqrt(9+1+4) = sqrt(14) = 3.74 gives the angular speed of that rotation.

A curl of zero means the field is irrotational. Conservative fields — gravity, electrostatics, ideal fluid potential flow — always have zero curl. This is precisely what allows them to be written as the gradient of a scalar potential.

Stokes’ theorem connects the curl integrated over a surface to the line integral around its boundary, and is the 3D generalization of Green’s theorem.