2.15 : Divergence and Stokes' Theorems

The divergence and Stokes' theorems are a variation of Green's theorem in a higher dimension. They are also a generalization of the fundamental theorem of calculus. The divergence theorem and Stokes' theorem are in a way similar to each other; The divergence theorem relates to the dot product of a vector, while Stokes' theorem relates to the curl of a vector. Many applications in physics and engineering make use of the divergence and Stokes' theorems, enabling us to write numerous physical laws in both integral form and differential form. Each theorem has an important implication in fluid dynamics and electromagnetism. Through the divergence theorem, a difficult surface integral can be transformed easily into a volume integral, and vice versa. The rate of flow or discharge of any material across a solid surface in a vector field, like electric flow, wind flow, etc., can be determined using the divergence theorem. Similarly, Stokes' theorem can be used to transform a difficult surface integral into an easier line integral, and vice versa. The line integral in itself can be evaluated using a simple surface with a boundary.