The differential calculus for functions which map one normed vector space into another normed vector space is the main prerequisite for the study of manifolds and Lagrangian mechanics in the modern idiom. The mild degree of abstraction involved helps focus attention on the simple geometric ideas underlying the basic concepts and results. The insights and techniques which this approach fosters turn out to be very worthwhile in the study of many topics in applied mathematics.

The idea of a function as an entity in its own right, independently of any numerical variables which may be used to define it, is fundamental for this approach to calculus. This idea, although usually confined to statements of the theory, can easily be used to provide computational tools for particular examples. This involves the development of a “variable free” language of calculus in which the functions themselves, as distinct from the values that they take are highlighted. This enables us to formulate and work with many important and difficult mathematical ideas in a simple manner. Some of the notation does not appear elsewhere but a good account of the basic ideas may be found in Spivak(1965) or in Lang(1964).

FRECHET DERIVATIVES

Here we outline how the idea of a derivative can be formulated for functions which map one normed vector space to another. The basic idea is this: a function is differentiable at a point if there is an affine map which approximates the function very closely. The derivative of the function at this point is then the linear part of this affine map.