divergence and curl

Drop a tiny imaginary speck into a flowing fluid and watch what happens around it. Two questions capture the local behaviour. First: is fluid, on balance, streaming out of this little spot or piling into it — is there a source or a drain here? That is divergence. Second: would a tiny paddlewheel placed at the spot start to spin, because the flow on one side is faster than on the other? That tendency to rotate locally is curl.

These are the two fundamental derivatives of a vector field F. The divergence is a number at each point, written div F or ∇·F = ∂P/∂x + ∂Q/∂y + ∂R/∂z; it is positive where the field spreads out (a source), negative where it converges (a sink), and zero where flow in matches flow out. The curl is itself a vector, written curl F or ∇×F, whose direction is the axis of local rotation and whose length measures how fast that swirling is. A field with zero curl everywhere is called irrotational.

Far from being abstract, these two operations are the heart of vector calculus and of physics. James Clerk Maxwell wrote the laws of electricity and magnetism almost entirely in the language of divergence and curl: electric charge is the divergence of the electric field, and a changing magnetic field gives the electric field a curl. They also govern fluid flow and heat transport, and they tie together the great integral theorems of Gauss and Stokes.