Guide to Mathematical Methods pp 175-201 | Cite as

# Functions of Two Variables

## Abstract

We have studied functions of one variable and vector functions of one variable. In each case, the value of the variable defines the value of the function. We now turn to functions of two variables, where the values of *two* variables must be given before we can evaluate the function. As in the case of a function of one variable, we define a function *f* of two variables, *x* and *y*, say, by its value at all points of the domain. Now, however, ranges of values for both *x* and *y* must be specified. For example, the domain of *f* might consist of values of *x* and *y* satisfying *a* ≤ *x* ≤ *b* and *c* ≤ *y* ≤ *d*. Instead of referring to *x* and *y* separately, we shall often refer to them as a point (*x, y*); then we can talk about *f* being evaluated at the point (*x*, *y*), while the domain of *f* is the rectangle in the *xy* plane defined by the above inequalities.

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