What Is Curl?
Where divergence asks whether a vector field spreads, curl asks whether it spins. Drop a tiny paddle wheel into the flow at a point: curl measures how fast it would rotate, and in which sense. For a plane field F = ⟨P, Q⟩ the (scalar) curl is:
curl F = ∂Q/∂x − ∂P/∂y
Positive means counterclockwise rotation, negative clockwise, zero an irrotational point, the paddle wheel drifts without turning. The formula's two terms detect the two ways spin arises: the vertical component Q strengthening left-to-right torques the wheel counterclockwise; the horizontal component P strengthening bottom-to-top torques it clockwise (hence the minus). In full 3D, curl is a vector (∇×F) with three such components; the plane case is its z-component, the only one that survives when the field is flat.
How to Use This Calculator
Enter both components of the field. The calculator computes ∂Q/∂x and ∂P/∂y symbolically, forms the difference, and, at your chosen point, reports the rotation rate with its sense (counterclockwise, clockwise, or irrotational). The arrow plot lets you check by eye: the rotation chip visibly swirls, the radial chip visibly doesn't, and the shear chip demonstrates the subtlest case, arrows all parallel yet curl nonzero, because a paddle wheel in shear does spin, its top pushed harder than its bottom.
Worked Example
The rigid-rotation field F = ⟨−y, x⟩, the velocity field of a turntable spinning counterclockwise. Partials: ∂Q/∂x = 1, ∂P/∂y = −1, so
curl F = 1 − (−1) = 2, everywhere
Uniform curl: every point of a rigid turntable rotates at the same angular rate (here, curl = twice the angular velocity, a general fact for rigid rotation). Contrast the fourth chip: ⟨2xy, x²⟩ has ∂Q/∂x = 2x and ∂P/∂y = 2x, curl ≡ 0. That field is the gradient of x²y, and gradients always have zero curl: conservative fields don't rotate, the observation at the heart of potential theory.
Curl Zero and Conservative Fields
The equality curl(∇f) = 0 has a powerful near-converse: on a domain without holes, a field with zero curl everywhere is a gradient field, it's conservative meaning line integrals depend only on endpoints and work done around any closed loop vanishes. The curl computation is therefore the standard screening test before hunting for a potential function: nonzero curl anywhere means no potential exists; zero curl (on a simply connected region) guarantees one does. This is exactly the exactness test for differential forms, and Green's theorem makes the loop-integral connection quantitative: circulation around a closed curve equals the double integral of curl over the enclosed region.
Common Mistakes to Avoid
- Swapping the subtraction. Curl is Qx − Py cross-derivatives, Q first. Reversing signs flips every rotation verdict; pairing straight (Px Qy) computes divergence instead.
- Concluding "no rotation" from straight arrows. The shear chip refutes it: parallel arrows of varying strength still spin the paddle wheel. Curl reads differential motion, not curved streamlines.
- Concluding "rotation" from curved streamlines. The reverse error also exists: the vortex field ⟨−y, x⟩/(x²+y²) has circular streamlines yet zero curl away from the origin, fluid circles the drain without locally spinning. Streamline shape and local rotation are independent.
- Skipping the domain check for conservativeness. Zero curl promises a potential only on hole-free regions; that same vortex field is the classic counterexample on the punctured plane.
Real-World Applications
Curl is the vorticity of fluid dynamics: weather maps of curl(wind) locate cyclones and anticyclones, hurricane intensity is tracked through vorticity concentration, and aircraft wingtip vortices, hazardous curl shed into the air, dictate takeoff spacing at every airport. Aerodynamic lift itself is circulation (curl integrated over area, via Green's theorem) bound around the wing: no curl, no flight, by the Kutta–Joukowski theorem.
Electromagnetism runs on the other two Maxwell equations: curl E = −∂B/∂t is Faraday's law, changing magnetic fields wrap electric fields around them, the operating principle of every generator and transformer, and curl B relates to currents, wrapping magnetic fields around wires (Ampère's law), which is why compasses circle a current-carrying cable. And the conservative-field screening test is daily practice in mechanics and thermodynamics: checking curl = 0 before defining potential energy, exactly as this calculator's fourth chip models.
Frequently Asked Questions
Why is 2D curl a number but 3D curl a vector?
Rotation in 3D needs an axis, curl's vector direction is that axis (right-hand rule), its magnitude the rotation rate. A plane field can only rotate about the out-of-plane axis, so only the z-component survives: the scalar this page computes is that lone component.
What's the paddle wheel picture, precisely?
An infinitesimal wheel with frictionless bearings, its axis out of the plane: the flow's differential push on its blades makes it spin at angular velocity curl/2. It's a genuine measurement device conceptually, vorticity meters in hydraulics labs work on the principle.
How do Green's and Stokes' theorems involve curl?
Green's (2D): circulation around a closed curve = double integral of scalar curl inside. Stokes' (3D): circulation around a loop = flux of curl through any surface it bounds. Both say "add microscopic spins to get macroscopic circulation", curl is circulation density, exactly as divergence is flux density.
Why do gradient fields always have zero curl?
Because curl(∇f) = fyx − fxy which vanishes by Clairaut's mixed-partials theorem. Conservative force fields (gravity, electrostatics) inherit their zero curl from being gradients of potentials, the mixed-partial equality wearing a physics costume.
Can a field have both curl and divergence nonzero?
Certainly, generic fields blend expansion and rotation (plus shear), and the two derivatives are independent measurements. Helmholtz's theorem makes the blend precise: any reasonable field decomposes into a curl-free part plus a divergence-free part, the mathematical basis of splitting flows into potential and vortex components.
What units does curl carry?
Field units per length, for a velocity field (m/s), curl comes out in 1/s: an angular frequency, consistent with the paddle wheel spinning at curl/2 radians per second. Tracking units through div and curl is a quick sanity check that the right derivative combination was taken.