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Functions of Several Variables - (12.1)
1. Functions of Several Variables
A function of two variables, say x and y, is a relation that assigns exactly one real number to
each ordered pair of real numbers x, y . A function of three variables, say x, y and z, is a relation
that assigns exactly one real number to each ordered pair of real numbers x, y, z . For example,
a. fx, y  2xy 2
b. fx, y, z  x y 2  sin y e z
The domain of a function in two variables x, y is a set that contains all ordered pairs x, y at which
f is defined. The range of a function is the set of real numbers which are images of the function. The
graph of a function of two variables is the graph of the equation z  fx, y which is a surface in
space. We are not able to graph a function of three variables.
Example Find and sketch the domain of the function.
ln y
a. fx, y 
b. Fx, y, z 
2x 2 − 1
xy
,
c. Gx, y, z  36 − 4x 2 − 9y 2 − z 2
2
xz
a. fx, y is defined if y  0, and 2x 2 − 1 ≠ 0,  x ≠  1 . So, the domain of f is
2
D
x, y : y  0 and x ≠  1
2
b. Fx, y, z is defined if x  y ≥ 0, x ≠ 0 and z ≠ 0. So, the domain of F is
D  x, y, z : x  y ≥ 0, x ≠ 0 and z ≠ 0
c. Gx, y, z is defined if 36 − 4x 2 − 9y 2 − z 2 ≥ 0, that is 36 ≥ 4x 2  9y 2  z 2 . So, the domain of G
is
D  x, y, z in the sphere of radius 6
Example Give the domain and range of the function. Sketch the graph of the function.
b. fx, y  sin x cos y
c. fx, y  e −x/2 sin y
a. fx, y  x 2  4y 2  1
a. Let z  x 2  4y 2  1. The domain of f is
D  x, y : −  x  , −   y  
and the range of f is
R  z : z ≥ 1.
Graph the following ellipses: z  1, z  5, z  17.
when z  1, x 2  4y 2  1  1  x 2  4y 2  0  x  0 and y  0, a point 0, 0.
2
when z  5, x 2  4y 2  1  5, x 2  4y 2  4  x  y 2  1, an ellipse with a  2, b  1
4
2
y2
when z  17, x 2  4y 2  1  17, x 2  4y 2  16  x 
 1, an ellipse with a  4, b  2
16
4
1
4
15
3
10
2
5
-2
-4
1
-2
-1
0v u0
2
-2
2
4
0
yv
-1
1
2
4
-4
-2
1
2
z  x 2  4y 2  1
– z  5, -.- z  17
b. Let z  sin x cos y. The domain of the function is
D  x, y : −  x  , −   y  
and the range of the function is
R  z :

, and x 
Graph the following curves: x 
6
When x   , z  sin  cos y  12 cosy.
6
6
When y  0, z  sinx cos 0  sinx.
−1 ≤ z ≤ 1
 , y  0.
2
When x   , z  sin 
2
2
1
1
0.5
-5
-5
0.5
-5
-1
5
5
z  sin x cos y
–x 

6
,
-1

2
and the domain of the function is
z : −  z  
Graph the following curves: x  0, and x  1, y   , and y   .
6
2
When x  0, z  e −0 sin y  siny. When x − 1, z  e −1/2 sin y.
R
5
, ... y  0,
c. Let z  e −x/2 sin y. The domain of the function is
D  x, y : −  x  , −   y  
2
-5
y t
-0.5
y x
-0.5
5
cos y  cosy.

3
When y   , z  e −x/2 sin 
6
6

1
2
e −x/2 and y   , z  e −x/2 sin 
2
2
 e −x/2
10
5
6
4 4 22
10
8
6
4
2
0
-2-2 -4-4 -6
-5
-5
4
2
-10
y
t
-4
-2
5
x
y
z  e −x/2 sin y
– x  0, 1, . . . y 

6
,

2
2. Level Curves and Contour Plots
A level curve of the function fx, y is a 2-D graph of the equation: fx, y  c, for some constant c
in the range of f. A contour plot of fx, y is a graph of many level curves fx, y  c for c in the
range of f.
For a function of 3 variables, say Fx, y, z the graph of the equation Fx, y, z  c is a surface which is
called a level surface.
Example Sketch the contour plots of fx, y.
a. fx, y  x 2  4y 2  1
b. fx, y  −x 2  y
a. Let z  x 2  4y 2  1. The range of z is: z ≥ 1. Let c ≥ 1. The level curves:
x 2  4y 2  1  c  x 2  4y 2  c − 1 a point or ellipses
c  1, x 2  4y 2  0, the level curve is a point 0, 0
2
c  5, x 2  4y 2  4 or x  y 2  1, an ellipse with a  2, b  1
4
4
2
-4
-2
0
-2
-4
3
2t
4
-4
8
6
-2
4
2
-2
y
u
1
-1
-3
2
-2
-1
1
-2
2
3
-4
v2
-6
-8
x
a. z  x 2  4y 2  1
b. Let z  −x 2  y. The range of z is:
b. z  −x 2  y
−,  . Let c be a real number. The level curves:
− x 2  y  c  y − c  x 2 parabolas
c  −2, y  x 2 − 2
c  0, y  x 2
c  2, y  x 2  2
15
10
5
-4
-2
0
2t
4
Example Sketch the level surfaces of Fx, y, z  4x 2  9y 2  z 2 for c  0, c  9, c  36.
Let w  Fx, y, z. The range of F is: w ≥ 0. For c ≥ 0, the level surface 4x 2  9y 2  z 2  c is an
ellipsoid or a point.
c  0, 4x 2  9y 2  z 2  0 a point 0, 0, 0
2
2
y2
c  9, 4x 2  9y 2  z 2  9 or x9 
 z  1, an ellipsoid with a  3 , b  1, c  3
2
1
9
4
2
2
y2
 z  1, an ellipsoid with a  3, b  2, c  6
c  36, 4x 2  9y 2  z 2  36 or x 
9
4
36
4
-3
5
2
-2
3
x2
9
4
-2
-1
-1
u v
1 1
-2 2
2

y2
1

z2
9
-6
-3
-4
6
3
x2
9
1
3. Density Plot
a. z  cosx 2  y 2 
b. z  cose x  e y 
-2
-2
u v
2 2
4
-5 4

y2
4

z2
36
-4
-6
6
1
d. z  e −xy
c. z  lnx 2  y 2 
1
1 -2
-1
0.5
y
0.5
-2
2
y
x
1
-0.5
-1
5
2
2
1
x
-1
-2
-2
-1
x
2 -2
-1
y1
1
0
1
2
20
15
2
-1
10
-3
5
-4
-2
0
yx
-1
2
-2
1
-2
-2
-1
-1
2
y
2
-2
-1
1
2
1
y
-1
1
-2
1
x
2
2
x
-2
-2
-1
-1
y
2
-1
1
-2
y
2
-1
1
1
x
2
6
x
-2