Calculus×Computing
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Curl

Level UniversityDifficulty ★★★★★Concept⌖ Open in the map

What is it?

∇×F\nabla\times F: a vector measuring the local rotation of a field — its axis is the axis of spin, its length twice the angular speed. Fields that are gradients have zero curl.

Formulas

∇×F=(∂yR−∂zQ, ∂zP−∂xR, ∂xQ−∂yP)\nabla\times F = \big(\partial_y R - \partial_z Q,\ \partial_z P - \partial_x R,\ \partial_x Q - \partial_y P\big)
∇×(∇ϕ)=0,∇⋅(∇×F)=0\nabla\times(\nabla\phi) = 0, \qquad \nabla\cdot(\nabla\times F) = 0
∇×E=−∂B∂t\nabla\times E = -\frac{\partial B}{\partial t}
Faraday's law

Where it shows up in computing

  • Fluid dynamics and CFD★★★★★fundamentalPhysics and simulation

    Vorticity ω=∇×u\omega = \nabla\times u describes eddies; "vorticity confinement" adds back swirls lost to numerical damping in film VFX.

  • Electromagnetism (Maxwell's equations)★★★★★fundamentalPhysics and simulation

    Faraday's and Ampère's laws are curl equations; induction in motors and antennas follows from them.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

⚛ Physics and simulation

3D Computer graphics

What depends on it

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.

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