Chapter 1 - Calculus on Euclidean Space

Abstract

This chapter focuses on the part of elementary calculus that deals with differentiation of functions of three variables and with curves in space, more specifically in Euclidean space. Euclidean 3-space, instead of saying that three numbers describe the position of a point, defines them to be a point. Elementary calculus does not always make a sharp distinction between the numbers and the functions. Indeed the analogous distinction on the real line may seem pedantic, but for higher-dimensional spaces such as R 3 , its absence leads to serious ambiguities. Here R 3 , or Euclidean 3-space, is the set of all ordered triples of real numbers. Such a triple is called a point of R 3 . In linear algebra, it is shown that R 3 is, in a natural way, a vector space over the real numbers. Using R 3 , the chapter highlights various definitions relating to tangents, curves, and vector fields, which is dualized to 1-forms and that, in turn, lead to arbitrary differential forms. The notions of curve and differentiable function is generalized to that of a mapping function F: R n → R m . Starting from the usual notion of the derivative of a real-valued function, the chapter constructed appropriate differentiation operations for objects such as the directional derivative of a function, the exterior derivative of a form, the velocity of a curve, and the tangent map of a mapping. These differentiation operations all exhibited in one form or another, the characteristic linear and Leibnizian properties of ordinary differentiation.