Integral theorems

Integral theorems connect local and global statements

Green, Gauss, and Stokes’ theorems say that a sum of local derivatives over a region equals a flux or circulation on its boundary. They let a conservation law be written at the scale of a point or a control volume.

Use one dimension as the prototype

The fundamental theorem gives ∫_a^b f′ dx = f(b) − f(a): interior changes telescope to boundary values. In higher dimensions, divergence theorem gives ∫_V ∇·u dV = ∮_S u·n dA, while Stokes gives ∫_S (∇×u)·n dA = ∮_C u·dl. The orientation of n and boundary follows the right-hand rule.

For u = (x,0,0) in a unit cube, divergence is 1, so volume integral is 1. Flux is x=1 face area 1 minus x=0 face flux 0, also 1.

Boundary regularity is an assumption

The field must be sufficiently smooth, and the surface orientation must be consistent. Singular sources require excluding the singularity or adding its distributional contribution.

Work it: Verify the divergence theorem for u = (x,y,z) on a sphere of radius R: compute volume integral 3·(4πR³/3) and surface flux u·n = R.