Vector calculus tracks fields with gradients, curls, and flux
The subject differentiates and integrates vector fields mainly in three-dimensional Euclidean space, and is sometimes equated with broader multivariable calculus. Physics and engineering lean on it for electromagnetism, gravity, and fluid flow. Gibbs and Heaviside shaped modern notation from quaternion theory near the end of the nineteenth century.
Vector calculus—also called vector analysis—studies differentiation and integration of vector fields, chiefly in three-dimensional Euclidean space. The phrase sometimes covers multivariable calculus more broadly, including partial derivatives and multiple integrals. It feeds differential geometry and partial differential equations and is stapled into physics and engineering descriptions of electromagnetic, gravitational, and fluid fields. Developed from quaternion theory by J. Willard Gibbs and Oliver Heaviside late in the nineteenth century, its standard notation was fixed by Gibbs and Edwin Bidwell Wilson in the 1901 Vector Analysis, though Newton and others had pioneered related ideas. Cross-product form does not extend cleanly to higher dimensions; geometric algebra offers an alternative.
Scalar fields assign a number smoothly to every point—temperature, fluid pressure, or spin-zero quantum fields such as the Higgs. Vector fields attach a vector to each point, drawable as arrows in the plane, modeling fluid velocity or force strength and direction. Advanced treatments distinguish pseudovectors and pseudoscalars that flip under orientation reversal; curl of a vector field is a pseudovector. Pointwise vector algebra includes the usual products plus triple products. Differential operators built from del (nabla) give gradient, divergence, and curl, plus Laplacian operators; Jacobian matrices help when domain and range are both multivariable.
Three classic theorems generalize the fundamental theorem of calculus: gradient, divergence (flux), and curl (Stokes) theorems; in two dimensions divergence and curl reduce to Green's theorem. Linear approximations replace a differentiable f(x, y) near (a, b) by the tangent plane using partial derivatives. Critical points of smooth multivariable functions are where the gradient vanishes; local max, min, or saddle behavior is read from Hessian eigenvalues. Fermat's theorem says local extrema of differentiable functions occur at critical points, so zeros of the gradient plus Hessian tests locate them in theory.
Euclidean three-space supplies a norm from an inner product, angles, orientation, a volume form, and the cross product. Gradient and divergence need the inner product; curl and cross product also need handedness. The same toolkit works on oriented Riemannian three-manifolds via tangent spaces. Grad and div generalize to other dimensions; curl and cross product do not as directly—only in dimensions 3 and 7 does curl of a vector field remain a vector field in the classical sense. Geometric algebra and differential forms clarify the hidden identifications: in forms language, grad, curl, and div are exterior derivatives of 0-, 1-, and 2-forms, and the classical theorems are Stokes' theorem special cases.
Source: Vector calculus
