ABSTRACT
The Fundamental Theorem of Calculus expresses a relationship between the derivative and the definite integral. When a function depends on more than one variable, the connection is still there, although it is less transparent. In this chapter, the author looks at some multivariable generalizations of the Fundamental Theorem of Calculus. Although line integrals were used by physicists in the eighteenth century, the mathematical development came through the use of complex numbers, and the study of paths in the complex plane. The author looks at integrals of functions defined on a surface. This will require a precise definition of a surface with some examples. The right hand side is an integral over the boundary of a region, and the left sides feature the integral over the whole region, the integrand being obtained from the one on the right side using derivatives.
