Vector Field Curl Calculator
The curl of a vector field captures its local rotation: how much a tiny paddle wheel placed in the field would spin, and about which axis. It is the cross product of the del operator with the field and is itself a vector at every point. This calculator computes the curl from the six partial derivatives that define it, all evaluated at your point of interest, so the result is exact and deterministic. Enter the derivatives of the field components and the tool returns the three curl components and the curl magnitude.
Curl formula
curl F = del x F
x-component = dFz/dy - dFy/dz
y-component = dFx/dz - dFz/dx
z-component = dFy/dx - dFx/dy
magnitude = sqrt(cx^2 + cy^2 + cz^2)
The curl is the formal cross product of the del operator with the field. Each component pairs two cross partial derivatives, and the magnitude is the length of the resulting curl vector.
Notes on the curl
- The six inputs are partial derivatives already evaluated at your chosen point.
- A field with zero curl everywhere is irrotational.
- On a simply connected region, an irrotational field is conservative.
- The direction of the curl vector is the local axis of rotation.
- The magnitude measures the strength of the local swirl.
Curl: frequently asked questions
What is the curl of a vector field?
The curl measures the local rotation or circulation of a vector field. At each point it is itself a vector whose direction is the axis of rotation and whose magnitude is twice the local angular speed of an infinitesimal paddle wheel placed in the field.
What is the formula for the curl?
For a field F = (Fx, Fy, Fz), the curl is the vector (dFz/dy - dFy/dz, dFx/dz - dFz/dx, dFy/dx - dFx/dy). It is the cross product of the del operator with the field.
What does a zero curl mean?
A vector field with zero curl everywhere is called irrotational. On a simply connected region an irrotational field is conservative, meaning it is the gradient of some scalar potential and its line integrals are path independent.
Why do I enter partial derivatives instead of a function?
The curl is defined entirely by the partial derivatives of the field components, evaluated at your point of interest. Supplying those six derivatives lets the calculator stay exact and deterministic without parsing arbitrary symbolic functions.
What is the physical meaning of curl magnitude?
The magnitude of the curl vector tells you how strongly the field rotates about its axis at that point. A larger magnitude means a faster local swirl; zero magnitude means no rotation at that point.
Official sources
- NIST Digital Library of Mathematical Functions: Vectors and vector-valued functions.
- NASA Glenn Research Center: Vector basics.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.