Multivariate Calculus - Bruce E. Shapiro

Lecture 8
Functions of Two Variables
In this section we will extend our definition of a function to allow for multiple
variables in the argument. Before we formally define a multivariate function, we
recall a few facts about real numbers and functions on real numbers. Recall our
original, general definition of a function:
A function f is a rule that associates two sets, in the sense that each
object x in the first set D is associated with a single object y in the
second set R, and we write
and
f : D ↦→ R
y = f(x).
We call D the domain of the function and R the range of the
function.
In terms of functions of a real variable, we made some notational conventions:
• The symbol R represents the set of real numbers.
• Any line, such as any of the coordinate axes, is equivalent to R because there
is a one-to-one relationship between the real numbers and the points on a line.
• For real valued functions, the both the domain D and range R are subsets of
the real numbers, and we write
so that
If, in fact, D = R = R, we write
D ⊂ R
R ⊂ R
f : (D ⊂ R) ↦→ (R ⊂ R)
f : R ↦→ R
51