Vector calculus

Vector calculus studies derivatives and integrals of scalar and vector fields. Its central operations include gradient, divergence, curl, and integration along curves and across surfaces.

Scalar field: A function that assigns a scalar value to each point in a domain. Gradients describe how a scalar field changes in space.
Gradient: A vector of partial derivatives pointing toward the steepest increase of a scalar field. It converts scalar-field change into a direction and rate of increase.
Maxwell's equations: Four equations describing how electric and magnetic fields are generated and how they change. Their differential forms use divergence and curl to state electromagnetic laws locally.
Differential forms: Antisymmetric tensor fields that provide a coordinate-independent language for differentiation and integration on manifolds. They generalize vector-calculus operations beyond ordinary Euclidean space.
Vector field: A function that assigns a vector to each point in a domain. Divergence, curl, and field integrals operate on vector fields.
Divergence: A scalar measure of a vector field's local net outflow from a point. It detects sources and sinks in a field.
Navier–Stokes equations: Partial differential equations governing the motion of fluids. They use vector derivatives to describe fluid acceleration, pressure, and viscosity.
Tensor calculus: A framework for manipulating tensors and their derivatives across coordinate systems and curved spaces. It extends vector-calculus ideas to settings where vectors alone are insufficient.
Multivariable calculus: Calculus of functions with multiple input variables, including partial derivatives and multiple integrals. Vector calculus extends multivariable calculus with geometric field operations.
Curl: A vector measure of a three-dimensional vector field's local rotation. It quantifies the field's tendency to circulate around a point.

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