Multivariable Calculus Theorems Paragraph

Principle

Integrating a derivative over a region equals integrating the original function over its boundary.

Fundamental Theorem of Calculus

In 1D, the Fundamental Theorem of Calculus integrates f’ over [a,b] and gives f(b) - f(a).

Fundamental Theorem for Line integrals

Along curves, the Fundamental Theorem for Line Integrals integrate ∇f along a path and gives the difference in f at the endpoints.

Green’s Theorem

Over planar regions, Green’s Theorem relates a line integral around a close curve to a double integral of scalar curl over the region.

Stokes’ Theorem

Over surfaces, Stokes’ Theorem relates the surface integral of curl to a line integral along the boundary curve.

Divergence Theorem

Over volumes, the Divergence Theorem relates the volume integral of divergence to flux through the boundary surface.