Lecture 2: Surface Integrals and Advanced Theorems

This lecture introduces surface integrals, extending integration from curves to two-dimensional surfaces. Two main types are covered: scalar surface integrals for functions like mass density, and vector surface integrals (flux) for fields like fluid flow. The concept of parameterized surfaces is presented, along with tangent vectors, normal vectors, and the surface area element. Special attention is given to surfaces given as graphs. Stokes' Theorem is introduced as a generalization of Green's Theorem, relating line integrals around space curves to surface integrals of curl over bounded surfaces. The Divergence Theorem relates flux through closed surfaces to volume integrals of divergence over enclosed regions. It is applied to compute flux and derive Archimedes' principle. The lecture concludes with a summary table of the three great theorems: Green's Theorem, Stokes' Theorem, and the Divergence Theorem.