The Divergence Theorem

Seal a region of flowing fluid in an imaginary balloon. How much fluid escapes through the balloon's skin each second? The Divergence Theorem gives a stunningly simple answer: exactly as much as is being created inside. Every source within the volume must, in the end, flow out through the surface. The flux through the boundary equals the divergence summed over the solid. This is the capstone — the 3D crown on Green's and Stokes' theorems.