Multivariate Calculus - Bruce E. Shapiro

LECTURE 8. FUNCTIONS OF TWO VARIABLES 53
Figure 8.1: Visualization of a multivariate function. In both figures, the domain
represents a set in the xy-plane, and the range is a subset of the z-axis. In the
top, the function is visualized as a mapping between sets. The arrows indicate the
mapping for four points in the domain that are mapped to three points in the range.
The bottom figure visualizes this as a mapping embedded in 3D space, with each
point z = f(x, y) represented as a 3-tuple (x, y, z) in space.
(a) Don’t divide by zero
(b) Don’t take the square root of a negative number.
Anything that remains is part of the natural domain of the function
“Exclude the impossible and what remains, however improbable, is the
solution.” [Sherlock Holmes]
Example 8.4 Find the natural domain of the function
f(x, y) = √ 16 − x 2 − y 2
Solution. Since we can’t take the square root of a negative number, we exclude
points where
16 − x 2 − y 2 < 0
Thus the natural domain is the set of all (x, y) where
16 − x 2 − y 2 ≥ 0,
Math 250, Fall 2006 Revised December 6, 2006.