In this chapter, we discuss the major integral theorems that are used to develop physical laws based on integrals of vector differential operations. The general theorems include the divergence theorem, the Stokes' theorem, and various lemmas such as the Green's lemma.

The divergence theorem is a very powerful tool in the development of several physical laws, especially those that involve conservation of physical properties. It connects volume integrals with surface integrals of fluxes of the property under consideration. In addition, the divergence theorem is also key to yielding several other integral theorems, including the Green's identities, some of which are used extensively in the development of finite element methods.

Stokes' theorem involves surface integrals and contour integrals. In particular, it relates curls of velocity fields with circulation integrals. In addition to its usefulness in developing physical laws, Stokes' theorem also offers a key criteria for path independence of line integrals inside a given region that can be determined to be simply connected. We discuss how to determine whether the regions are simply connected in Section 5.3.

In Section 5.5, we discuss the Leibnitz theorems involving the derivative of volume integrals in both 1D and 3D space with respect to a parameter α in which the boundaries and integrands are dependent on the same parameter α These are important when dealing with time-dependent volume integrals.