vector identities

Once you have three differential operators — gradient, divergence, and curl — built from one symbol nabla, you find that combining them is not a free-for-all: certain combinations always vanish, and certain products always expand the same way. Vector identities are this fixed set of algebraic rules that the operators obey, the analog of the product rule and chain rule for vector fields. Knowing them turns long, error-prone component computations into one-line manipulations.

Two identities are so important they have names of their own. First, curl of a gradient is always zero: curl(grad f) = 0, for any scalar f. Second, divergence of a curl is always zero: div(curl F) = 0, for any vector field F. Both follow from the equality of mixed second partial derivatives and are the reason gradients are irrotational and curls are solenoidal. Beyond these, the product rules unwind compound expressions: grad(f g) = f grad g + g grad f; div(f F) = f div F + F dot grad f; curl(f F) = f curl F + (grad f) cross F; and the key second-derivative identity curl(curl F) = grad(div F) - nabla^2 F, which defines the vector Laplacian.

These identities are the working grammar of electromagnetism and fluid dynamics. The fact that div B = 0 lets you write B as the curl of a vector potential A (since div(curl A) = 0 automatically); the fact that curl E = 0 in statics lets you write E as minus the gradient of a scalar potential (since curl(grad phi) = 0). Deriving the electromagnetic wave equation from Maxwell's equations is essentially one application of curl(curl E) = grad(div E) - nabla^2 E. Memorizing a dozen identities saves you from re-deriving them in coordinates every time.