ABSTRACT
This chapter reviews some ideas from multi-variable calculus. It particularly emphasizes line and surface integrals and their interpretations as work and flux. The chapter develops the ideas in terms of both vector fields and differential forms, including the major theorems of vector analysis. It discusses Maxwell's equations from both points of view and our use of differential forms goes well beyond a typical multi-variable calculus course. The reader is surely familiar with the notions of Euclidean dot product and length in ℝn. The chapter assumes that functions are continuously differentiable; all partial derivatives are continuous functions. Even on the real line, there exist differentiable functions that are not continuously differentiable. The chapter mentions a fundamental result concerning the equality of mixed partial derivatives. It uses the notation of subscripts for partial derivatives for the first time; this notation is efficient and convenient.
