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.

Its magnitude is worth being precise about, because a lot of textbooks are casual here. The curl is twice the angular speed of a tiny paddle wheel dropped in the flow, not equal to it. For a rigid rotation with angular velocity ω the velocity field is v = ω × r, and curl v works out to 2ω. So a curl of magnitude 6 means the paddle wheel turns at 3 radians per second. In fluid dynamics the curl of the velocity field gets its own name, vorticity, precisely so nobody has to keep saying “twice the angular velocity”.

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, which 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 along the rotation axis. Its magnitude is sqrt(9+1+4) = sqrt(14) = 3.74, so a paddle wheel in this field spins at 3.74 / 2 = 1.87 radians per second about that axis.

A curl of zero means the field is irrotational. Conservative fields, meaning gravity, electrostatics and ideal potential flow, always have zero curl. That 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.

Curl is a cross product in disguise. Written as curl F = ∇ × F, the del operator sits in the first row of the same 3×3 determinant the cross product calculator expands, with partial derivatives standing in for components. Every sign in the formula above comes from that determinant, which is why the middle component looks backwards: it is the cofactor expansion, not a typo.