Calculus: Analytic Geometry and Calculus, with Vectors

576 Partial derivatives

11.3 Formulas involving partial derivatives

problems require us to learn more about partial derivatives and some

formulas that have important applications. The first part of the section

is a rather dismal discussion of unlovely terminology designed to promote

understanding of curly dee abbreviations. To begin, let f be a function

of two "secondary variables" x and y, and let gl and g2 be functions of a

single "primary variable" a. Then, in appropriate circumstances, we

can set x = gl(a), y = g2(a), and define a function F of the single pri-

mary variable a by the formula

(11.31) F(a) = f(x,Y) = f(gl(a), g2(-))-

In this and all similar situations in this section, we suppose that the

arguments of functions of one "variable" are confined to intervals over

which the functions are differentiable and that the arguments of functions

of more than one "variable" are confined to regions over which the func-

tions have continuous partial derivatives of first order. Differentiating

(11.31) with the aid of the chain rule then gives a result that can be

written in ways that look very different. Using notation of one brand gives

F'(a) = f=(gl(a), g2(a))gi(a) + f,,(gi(a), g2(a))g2(a)-

This can be put in the form

(11.321) F'(a) = f=(x,Y)gi(a) + fv(x,y)g2(a)

and we are responsible for remembering that the secondary variables

are linked to the primary variable by the formulas x = gl(a), y = g2(a)-

Next, we can put this in the form

(11.322) F'(«)=axda+ayd«

Finally, as we usually do when the numbers in (11.31) represent tempera-

ture or something having recognizable significance, we denote the mem-

bers of (11.31) by a single appropriately chosen letter, say u, and write

(11.33) du

au dx+au dv

da

_

ax da ay da

The abbreviated formula (11.33) is now expected to tell us that u is

linked to a primary variable a by the secondary varables x and y that

are identified by the fact that au/ax and au/ay appear in the formula.

This formula makes sense and enables us to make calculations when, for

example, we have the formulas u = x2 + y, x = cos a, y = sin a.

One application of this material is worthy of mention. Suppose that

a function F(x,y) of two "secondary variables" and a constant c are given

and that y(x) is a function of the "primary variable" x such that

(11.34) F(x,y) = c