15.1: Functions of Several
Variables
Overview:
! Real valued functions f : R ! R
(Differential, Integral Calculus)
! Vector Valued functions f : R ! R n
Space Curves, Ch. 14, Skip for now
! Functions of Several Variables
f : R n ! R, Chapter 15
! Vector valued functions of several
variables
Vector fields, Chapter 17
1
Terminology:
! function of two variables f : R 2 ! R
f!x, y" " z
! domain, range
! independent variables, dependent
variables
Extra Problems for 15.1
8, 9, 13, 17, 23, 29, 37, 39, 59
2
Examples:
f!x, y" " cos!y"e sin x
Def: If f is a function of two variables
with domain D, then the graph of f is
#!x, y, z" " R 3 | z " f!x, y" $
for !x, y" " D.
0
4
2
y 0
-2
-4
-4
-2
0x
4
2
Def: The level curves of a function
f!x, y" are the curves with equations
f!x, y" " k where k is a constant in the
range of f.
f!x, y" " 4 ! x 2 ! y 2
0
-5
-2
-2
y 0
0 x
2
2
3
4
Examples:
Functions of 3 or more variables
f!x, y, z"
f!x 1 , x 2 , %x n "
0
4
0
y
4
2
y 0
-2
-4
-4
-2
0 x
2
4
Level surfaces
2
-2
4
x
-4
-4
4
2
0
-3
-2
-3
-3
-2
-4
-2
-1
y 0
1
2
3
3
2
1
-1
0 x
-2
-3
-2
-1
y 0
-1
0 x
1
1
2
2
3
3
5
6