Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

Sec. 1. Faiic Notions
Chapter VI
FUNCTIONS OF SEVERAL VARIABLES
1. The concept of a function of several variables. Functional notation.
A variable quantity 2 is called a single-valued function of two variables jc,
y, if to each set of their values (x, //) in a givm range there corresponds a
unique value of z The variables x and y are called arguments or independent
variables. The functional relation is denoted by
* = /(*, y).
Similarly, we define functions of three or more arguments.
Fxample 1. Express the volume of a cone V as a function of its generatrix
x and of its base radius y
Solution. From geometry we know that the volume of a cone is
where h is the altitude of the cone. But h y ** y 2 - Hence,
Fig. 63
This is the desired functional relation.
The value of the function z^f(x.y) at a
point P (a.b). that is, when x=^-a and y b t
is denoted by / (a,b) or f (P) Generally speaking,
the geometric representation of a function
like z f (x,y) in a rectangular coordinate
system X, Y. Z is a surface (Fig. 63).
Solution. Substituting r=2 and t/= 3, we find
Example 2. Find/ (2, 3) and/1, if