mathematics//calculus//vector calculus

Vector calculus is calculus on scalar and vector fields: the operators built from the differential operator nabla (\(\nabla\)), and what each of them measures locally.

Vector calculus is calculus on scalar and vector fields: the operators built from the differential operator nabla (∇\nabla∇), and what each of them measures locally.

The note on gradient, divergence and curl separates the three uses of the symbol and the sign conventions that go with them.

It is the part of calculus that an engineer meets in fields spread over space: the temperature along a motor winding, the velocity of air over a propeller, the potential around a power line. Two of its matrices are used far beyond fields. The Jacobian collects the first partial derivatives of a vector function and is what linearization, the extended Kalman filter and uncertainty propagation compute; the Hessian collects the second derivatives of a scalar function and measures the curvature an optimizer has to cross. A field equation written with these operators is a PDE, which in practice is cut into nodes and simulated as a large system of ODEs.