Technical Mathematics, Sixth Edition

17.6. Functions of More Than One Variable

So far in this chapter, we have considered only cases where y was a function of a single variable x. In functional notation, this is represented by y = f(x). In this section we cover functions of two or more variables, such as

y = f(x, w)

and

y = f(x, w, z)

and so forth.

17.6.1. Joint Variation

When y varies directly as both x and w, we say that y varies jointly as x and w. The three variables are related by the following equation, where, as before, k is a constant of proportionality:

NOTE

APPLICATION: The expansion of a pipe varies jointly with the temperature change and the initial length. If a 5.00-in. pipe expands 0.00225 in. with a 15° temperature increase, by how much will the pipe expand with a 25° temperature increase? ANSWER: 0.00375 in.

Example 32: If y varies jointly as both x and w, how will y change when x is doubled and w is one-fourth of its original value? Solution: Let y′ be the new value of y obtained when x is replaced by 2x and w is replaced by w/4, while the constant of proportionality k, of course, does not change. Substituting in Eq. 51, we obtain But, since kxw = y, then So the new y is half as large its former value.

17.6.2. Combined Variation

Relationships more complicated than that in Eq. 51 are usually ...