Differentiable Functions of Several Variables

CHAPTER 16. Differentiable Functions of Several Variables The Differential and Partial Derivatives Let w = f (x; y; z) be a function of the three Variables x; y; z. In this chapter we shall explore how to evaluate the change in w near a point (x0 ; y0 ; z0 ), and make use of that evaluation. For Functions of one variable, this led to the derivative: dw=dx is the rate of change of w with respect to x. But in more than one variable, the lack of a unique independent variable makes this more complicated. In particular, the rates of change may differ, depending upon the direction in which we move. We start by using the one variable theory to define change in w with respect to one variable at a time. Definition Suppose we are given a function w = f (x; y; z). The partial derivative of f with respect to x is defined by differentiating f with respect to x, considering y and z as being held constant.

This leads to the following restatement of the definition of differentiability: Proposition 16.1 Suppose that w = f (x; y z) is differentiable at x0 y0 z0. Then (16.23) dw = ∂f ∂x dx + ∂f ∂y dy ∂f ∂z dz: There are a variety of ways to use formula (16.23), which we now illustrate. Example 16.6 Let (16.24) z = f (x; y) x2 xy + y3:

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