Divergence Theory and Surface Integrals

This lecture delves into the application of the divergence theory to compute the volume of a three-dimensional unit ball using surface integrals. The instructor demonstrates the transformation of the integral using spherical coordinates, simplifying the calculations step by step. The divergence theorem is then applied to relate the volume integral to the surface integral, resulting in the derivation of the volume of the 3D unit ball. Additionally, a second example involving a cylinder with three distinct surface pieces is explored, showcasing the complexity that can arise in engineering applications. Parametrizations and cross-products are introduced to facilitate the computation of surface integrals over the different components of the cylinder.