Curl

The curl of a vector field F, written ∇×F, is itself a vector field that measures how much F circulates around each point. Imagine dropping a tiny paddle-wheel into a fluid whose velocity is F: the curl at the paddle-wheel's location tells you which axis the wheel will spin around (the direction of ∇×F) and how fast (the magnitude). Where the curl is zero, the wheel sits still; where the curl is nonzero, the wheel spins, and the field has rotational structure.

Mathematically, ∇×F is a determinant-style construction with components (∂F_z/∂y − ∂F_y/∂z, ∂F_x/∂z − ∂F_z/∂x, ∂F_y/∂x − ∂F_x/∂y). Stokes' theorem connects local curl to a closed-loop integral: the flux of ∇×F through any open surface equals the line integral of F around the boundary loop, ∫(∇×F)·dA = ∮F·dℓ. This is the calculus that makes Ampère's law sing: in differential form ∇×B = μ₀J + (1/c²)∂E/∂t says "the curl of the magnetic field at every point equals μ₀ times the local current density plus a displacement-current term", and Stokes' theorem turns that into the integral statement ∮B·dℓ = μ₀ I_enc that's easier to apply.

Curl is one of two fundamental vector-calculus operations that decompose any vector field, the other being divergence. Helmholtz's theorem says any well-behaved vector field is uniquely determined (up to a constant) by its divergence and its curl together — so once you know ρ = ε₀∇·E and J = (1/μ₀)∇×B (plus the displacement-current piece) for the electromagnetic field, you have specified the field completely. Maxwell's equations are exactly the four divergence and curl equations needed to do this for E and B simultaneously.