Green's Theorem

Definition

Green's theorem relates a double integral over a region in the plane to a line integral around its boundary. It establishes an equivalence between these two types of integrals.

Analogy

Think of Green's theorem as connecting two worlds - one world where you calculate the area inside a region, and another world where you measure the effect along its boundary. Green's theorem is like a bridge that allows you to travel between these two worlds.

Related terms

Divergence Theorem: The divergence theorem relates a triple integral over a region in space to a surface integral over its boundary. It extends the idea of Green's theorem into three dimensions.

Stokes' Theorem: Stokes' theorem is an extension of Green's theorem to higher dimensions. It relates the curl of a vector field over a surface to the line integral around its boundary.

Flux: Flux measures how much of something (such as fluid or energy) flows through a given surface. It can be calculated using either Green's theorem or the divergence theorem.