ABSTRACT
In elementary courses in mathematics, functions are often thought of as things which have a formula associated with them and it is the formula which receives the most attention. For example, in beginning calculus courses the derivative of a function is defined as the limit of a difference quotient. We start with one function which we tend to identify with a formula and, by taking a limit, we get another formula for the derivative. A jump in abstraction occurs as soon as we encounter the derivative of a function of n variables where the derivative is defined as a certain linear transformation which is determined not by a formula but by what it does to vectors. When this is understood, we see that it reduces to the usual idea in one dimension. The idea of weak partial derivatives goes further in the direction of defining something in terms of what it does rather than by a formula, and extra generality is obtained when it is used. In particular, it is possible to differentiate almost anything if we use a weak enough notion of what we mean by the derivative. This has the advantage of letting us talk about a weak partial derivative of a function without having to agonize over the important question of existence but it has the disadvantage of not allowing us to say very much about this weak partial derivative. Nevertheless, it is the idea of weak partial derivatives which makes it possible to use functional analytic techniques in the study of partial differential equations and we will show in this chapter that the concept of weak derivative is useful for unifying the discussion of some very important theorems. We will also show that certain things we wish were true, such as the equality of mixed partial derivatives, are true within the context of weak derivatives.
