-
Chapter 3
Functions of SeveralVariables
3.1 Definitions and Examples of Functions oftwo or More
Variables
In this section, we extend the definition of a function of one
variable to functionsof two or more variables. You will recall that
a function is a rule which assignsa unique output value to each
input value. It is similar for functions of two ormore variables.
The only difference is that the input is not a number anymore,it is
a pair, a triple, ... The output will be a real number. In other
words,in this chapter, we will be dealing with functions of the
form f : R2 → R orf : R3 → R. Here is a more formal definition.
Definition 3.1.1 Let D = {(x, y) : x ∈ R and y ∈ R} be a subset
of R2.
1. A real-valued function of two variables f : D → R is a rule
which assignsto each ordered pair (x, y) in D a unique real number
denoted f (x, y).
2. The set D is called the domain of f . Usually, when defining
a function,one must also specify its domain. When the domain is not
specified, it isunderstood that the domain is the largest possible
set of input values thatis the set of values of x and y for which f
(x, y) is defined.
3. The set {f (x, y) : (x, y) ∈ D} is called the range of f . In
other words,the range if the set of output values.
4. Similarly, a real-valued function of three variables is a
rule which assignsto each triple (x, y, z) a unique real number
denoted f (x, y, z). As above,we have f : D → R but this time, D ⊆
R3. We can extend this definitionto as many variables as we
wish.
203
-
204 CHAPTER 3. FUNCTIONS OF SEVERAL VARIABLES
Example 3.1.2 Find the domain of f (x, y) =sin(x2 + y2
)x2 + y2
x2 + y2 is always defined, therefore sin(x2 + y2
)is always defined. Since x2 +
y2 6= 0 except when x = y = 0, it follows that f is always
defined except at(0, 0). So, its domain is R2 \ {(0, 0)}.
Remark 3.1.3 You will notice that the domain is not a set of
values. Rather,it is a set of pairs.
Example 3.1.4 Find the domain of g (x, y) =sinx cos y
x− yThe numerator is always defined, so is the denominator.
However, the de-nominator cannot be zero. It is zero when y = x.
The domain is the setR2� {(x, x) : x ∈ R}. We could also write that
the domain is
{(x, y) ∈ R2 : y 6= x
}.
Example 3.1.5 Find the domain of h (x, y) = x ln(y − x2
)ln is defined when its argument is positive. So, we see that
for h to be defined,we must have
y − x2 > 0y > x2
So, the domain of h is the portion of the xy-plane inside the
parabola y =x2. It is the yellow region in figure 3.1. We could
mwrite that the domain is{
(x, y) ∈ R2 : y > x2}.
3.1.1 Closed and Bounded Sets
In this section, we extend to two and higher dimensions the
notion of closedand open intervals. You will recall that a closed
interval on the real line is aninterval which contains its
endpoints. So, [a, b] is a closed interval, but [a, b),(a, b] and
(a, b) are not closed. There is a similar notion for subsets of R2
andR3. In this section, we will not present this material very
thoroughly. This isusually done in an advanced calculus or in a
real analysis class. The intent hereis to give the reader an idea
of what the notion of closed set in R2 and R3 is.
Definition 3.1.6 (boundary point) We extend the notion of the
end pointof an interval to higher dimensions. Such points are
called boundary points.
1. Let D be a subset of R2. A boundary point of D is a point (a,
b) suchthat every disk centered at (a, b) contains both points of D
and points notin D.
2. Let D be a subset of R3. A boundary point of D is a point (a,
b, c) suchthat every sphere centered at (a, b, c) contains both
points of D and pointsnot in D.
Definition 3.1.7 (interior point) We give two definitions of an
interior point.
-
3.1. DEFINITIONS AND EXAMPLES OF FUNCTIONS OF TWOORMORE
VARIABLES205
Figure 3.1: Domain of h (x, y) = x ln(y − x2
)
-
206 CHAPTER 3. FUNCTIONS OF SEVERAL VARIABLES
1. Let D be a subset of R2 or R3. An interior point of D is a
point in Dwhich is not on the boundary of D. The set of interior
points of a givenset is called the interior of that set.
2. Let D be a subset of R2. A point P of D is an interior point
of D if thereexists a disk containing P which is included in D.
3. Let D be a subset of R3. A point P of D is an interior point
of D if thereexists a sphere containing P which is included in
D.
Remark 3.1.8 Let us make the following remarks:
1. This agrees with our intuitive definition of a boundary. If
you were on theboundary between two countries, stepping on one side
would put you in onecountry, stepping on the other side would put
you in the other country.Every disk around you would include parts
of both countries.
2. The definition of a boundary point does not require the
boundary point ofa set be in the set. We will see in the examples
the boundary points ofa set are not always in the set. In fact, it
is a special property a set haswhen it contains all its boundary
points.
3. An interior point of a set is always in the set.
4. An interval on the real line is the equivalent of a disk in
the plane anda sphere in space. They represent a region around a
point. When we donot specify the dimension, we will use the term
ball. Thus a ball can bean interval, a disk or a sphere, depending
on which dimension we are in.The term ball is also used in higher
dimensions.
Example 3.1.9 The boundary of the disk defined by x2 + y2 ≤ 1 is
the circlex2 + y2 = 1. Its interior is the disk x2 + y2 < 1.
Example 3.1.10 The boundary of the disk defined by x2 + y2 <
1 is the circlex2 + y2 = 1. Its interior is the disk x2 + y2 <
1.
You will note that the above two sets are different, yet they
have the sameboundary. The main difference is that the first set
contains its boundary, thesecond does not. This is an important
fact to remember. A boundary point ofa set does not necessarily
belong to the set.
Definition 3.1.11 (closed set) We extend the notion of a closed
interval tohigher dimensions. Let D be a subset of R2 or R3. D is
said to be closed if itcontains all its boundary points.
Definition 3.1.12 (open set) We extend the notion of an open
interval tohigher dimensions. Let D be a subset of R2 or R3. D is
said to be open if everypoint of D is an interior point of D.
-
3.1. DEFINITIONS AND EXAMPLES OF FUNCTIONS OF TWOORMORE
VARIABLES207
Figure 3.2: A Closed Set
Example 3.1.13 The disk defined by x2 + y2 ≤ 1 is closed because
it containsits boundary, the circle x2 + y2 = 1. Even if one point
from the boundary wereomitted, the set would no longer be
closed.
Example 3.1.14 The disk defined by x2 + y2 < 1 is open. Every
point is aninterior point.
Remark 3.1.15 When a set is closed, we represent its boundary
with a solidline as shown in figure 3.2. When it is open, we
represent its boundary with adashed line as shown in figure
3.3.
Definition 3.1.16 (bounded set) Let D be a subset of R2. D is
said to bebounded if it is contained within some disk of finite
radius. A subset D of R3is said to be bounded if it is contained
within some sphere of finite radius. Ingeneral, a subset D of Rn is
said to be bounded if it is contained within a ballof finite
radius.
Intuitively, this means a bounded set has finite extent.
Example 3.1.17 The disk defined by x2 + y2 ≤ 1 is bounded, it is
contained inany disk centered at the origin with radius larger than
one.
Example 3.1.18 The set{
(x, y) ∈ R2 : −2 ≤ x ≤ 2}is not bounded. It ex-
tends to infinity in the y-direction.
We will see that sets which are both closed and bounded have an
importantproperty related to finding extreme values later on in the
chapter.
-
208 CHAPTER 3. FUNCTIONS OF SEVERAL VARIABLES
Figure 3.3: An Open Set
3.1.2 Graphs of Functions of two Variables
Given a function of two variables f (x, y), for each value of
(x, y) in the domainof f , we can plot the point (x, y, z) where z
= f (x, y). The set of points inspace we obtain is called the graph
of f .
Definition 3.1.19 The graph of a function of two variables f (x,
y) is the setof points in space {(x, y, z) : (x, y) is in the
domain of f and z = f (x, y)}.
Like in 2-D, the 3-D graph of a function of two variables is
very helpful inthe sense that it helps to visualize the behavior of
f . The graph of a functionof two variables is a surface in space.
Unfortunately, graphing a function of twovariables is far more
diffi cult than a function of one variable. Fortunately forus, we
have technology which facilitates this task. Though we will not
spenda lot of time graphing functions of two variables, we will
explore some of theissues involved.We already know some simple 3-D
surfaces. For example, we saw that the
equation of a plane in space was of the form ax+ by + cz + d =
0. If c 6= 0, wecan solve for z and rewrite the plane as a function
of two variables.
Example 3.1.20 Find the function f (x, y) so that the plane
2x+3y−z+2 = 0can be written as z = f (x, y). Sketch its graph using
technology.We simply solve for z to obtain
z = 2x+ 3y + 2
-
3.1. DEFINITIONS AND EXAMPLES OF FUNCTIONS OF TWOORMORE
VARIABLES209
Figure 3.4: Plane 2x+ 3y − z = 2
Thus, we havez = f (x, y) = 2x+ 3y + 2
The graph of this function is shown on figure 3.4.
Remark 3.1.21 If we cannot solve for z as we did above, we can
still graph thecorresponding function using an implicit graph. many
graphing programs havethe capability of generating implicit
graphs.
If the graph is fairly simple, finding its intersection with the
coordinateplanes can be useful to help us visualize it. This is
done by:
1. To find the intersection with the xy-plane, set z = 0 in the
equation ofthe plane.
2. To find the intersection with the yz-plane, set x = 0 in the
equation ofthe plane.
3. To find the intersection with the xz-plane, set y = 0 in the
equation ofthe plane.
In the example above, the equation of the plane was 2x + 3y − z
+ 2 = 0.It intersects the xy-plane in the line 2x + 3y + 2 = 0, the
yz-plane in the line3y − z = 0 and the xz-plane in the line 2x− z =
0.
When a surface is more complicated to visualize, we do not limit
ourselvesto finding how it intersects the coordinate planes. We
look how it intersects anyplane parallel to one of the coordinate
axes. The curve we obtain are called thetraces or cross-sections of
the surface.
Definition 3.1.22 The traces or cross-sections of a surface z =
f (x, y) arethe intersection of that surface with planes parallel
to the coordinate planes,that is planes of the form x = C1, y = C2,
z = C3 where C1, C2 and C3 areconstants.
-
210 CHAPTER 3. FUNCTIONS OF SEVERAL VARIABLES
Figure 3.5: Topographic map of Kennesaw Mountain
Definition 3.1.23 The curves obtained by finding the
intersection of a surfacez = f (x, y) with planes parallel to the
xy-plane are also called contour curves.The projection of these
curves onto the xy-plane are called level curves. A plotmade of the
contour curve is called a contour plot.
The level curves are curves where the z value is constant. Level
curves areuses for example in mapping, to indicate the altitude.
The altitude is the sameeverywhere on a level curve. Figure 3.5
shows a topographic map of KennesawMountain, one can clearly see
the level curves indicating where the mountainis. Figure 3.6 shows
a 3-D rendering of the same area. On weather maps, levelcurves
represent isobars, that is areas where the atmospheric pressure is
thesame.
Example 3.1.24 Consider the surface z = f (x, y) = x2 + y2. Its
level curvesare of the form x2 + y2 = C, they are circles. It also
intersects planes parallelto the xz-planes in the curves z = x2,
which is a parabola. It intersects planesparallel to the yz-plane
in the curves z = y2, which is also a parabola.
3.1.3 Defining Functions of two Variables in Maple
We show the syntax by using an example. Suppose that we wish to
define
f (x, y) = sin(x2 + y2
)We would use the following syntax:
f:=(x,y)->sin(x^2+y^2);
-
3.1. DEFINITIONS AND EXAMPLES OF FUNCTIONS OF TWOORMORE
VARIABLES211
Figure 3.6: 3-D map of Kennesaw Mountain
Once defined, the user can evaluate the function for specific
values of x and ysimply by typing f (2, 3) for example. One can
also use the function name toplot it, or do other manipulations.To
plot z = f (x, y), use plot3d. See help in Maple for the correct
syntax.
To see a contour plot, use contourplot3d. Again, see help in
Maple for thecorrect syntax. If z is not defined explicitly in
terms of x and y. one must useimplicitplot3d (see Maple help for
the correct syntax).
3.1.4 Things to know
• Know what a function of two, three or more variables is.
• Be able to find the domain of such functions.
• Know what the level curves (surfaces) of such functions
are.
3.1.5 Problems
Make sure you have read, studied and understood what was done
above beforeattempting the problems.
1. What is the boundary of the set{
(x, y) ∈ R2 : 1 < x2 + y2 < 4}?
2. What is the boundary of the set{
(x, y) ∈ R2 : 1 ≤ x2 + y2 ≤ 4}?
3. Find the domain, range, level curves, boundary of the domain,
determineif the domain is open, closed or neither and determine if
the domain isbounded or unbounded for f (x, y) = y − x.
-
212 CHAPTER 3. FUNCTIONS OF SEVERAL VARIABLES
4. Find the domain, range, level curves, boundary of the domain,
determineif the domain is open, closed or neither and determine if
the domain isbounded or unbounded for f (x, y) = 4x2 + 9y2.
5. Find the domain, range, level curves, boundary of the domain,
determineif the domain is open, closed or neither and determine if
the domain isbounded or unbounded for f (x, y) = xy.
6. Find the domain, range, level curves, boundary of the domain,
determineif the domain is open, closed or neither and determine if
the domain isbounded or unbounded for f (x, y) = 1√
16−x2−y2.
7. Find the domain, range, level curves, boundary of the domain,
determineif the domain is open, closed or neither and determine if
the domain isbounded or unbounded for f (x, y) = ln
(x2 + y2
).
8. For each function below, sketch its graph, find and sketch
its level curves.
(a) f (x, y) = x2 + 2y2
(b) f (x, y) = x2 − y2
(c) f (x, y) = sin(x2 + y2
)(d) f (x, y) =
1
x2 + y2
9. Find an equation of the level curve of f (x, y) = 16− x2 − y2
through thepoint
(2√
2,√
2).
10. Find an equation of the level curve of f (x, y) =∫ yx
dt1+t2 through the point(
−√
2,√
2).
11. Find and sketch a typical level surface for f (x, y, z) = x+
z.
12. Find and sketch a typical level surface for f (x, y, z) = z
− x2 − y2
3.1.6 Answers
1. What is the boundary of the set{
(x, y) ∈ R2 : 1 < x2 + y2 < 4}?
There are two boundaries: the circle x2 +y2 = 1 and the circle
x2 +y2 = 4.
2. What is the boundary of the set{
(x, y) ∈ R2 : 1 ≤ x2 + y2 ≤ 4}?
There are two boundaries: the circle x2 +y2 = 1 and the circle
x2 +y2 = 4.
3. Find the domain, range, level curves, boundary of the domain,
determineif the domain is open, closed or neither and determine if
the domain isbounded or unbounded for f (x, y) = y − x.
(a) Domain: R2
(b) Range: R
-
3.1. DEFINITIONS AND EXAMPLES OF FUNCTIONS OF TWOORMORE
VARIABLES213
(c) Level curves: They are the curves y−x = c so it is the lines
y−x = c.(d) Domain boundary: None
(e) Domain open, closed or neither?: Both
(f) Domain bounded or unbounded?: Unbounded
4. Find the domain, range, level curves, boundary of the domain,
determineif the domain is open, closed or neither and determine if
the domain isbounded or unbounded for f (x, y) = 4x2 + 9y2
(a) Domain: R2
(b) Range: [0,∞)(c) Level curves: They are the curves in the
xy-plane 4x2 + 9y2 = c so
these are ellipses.
(d) Domain boundary: None
(e) Domain open, closed or neither?: Both
(f) Domain bounded or unbounded?: Unbounded
5. Find the domain, range, level curves, boundary of the domain,
determineif the domain is open, closed or neither and determine if
the domain isbounded or unbounded for f (x, y) = xy
(a) Domain: R2
(b) Range: R(c) Level curves: They are the curves y = cx(d)
Domain boundary: None
(e) Domain open, closed or neither?: Both
(f) Domain bounded or unbounded?: Unbounded
6. Find the domain, range, level curves, boundary of the domain,
determineif the domain is open, closed or neither and determine if
the domain isbounded or unbounded for f (x, y) = 1√
16−x2−y2
(a) Domain: The set of points satisfying 16−x2−y2 > 0 that is
x2+y2 <16 so it is the inside of the disk of radius 4, centered
at the origin.
(b) Range:[
14 ,∞
)(c) Level curves: They are the curves 16−x2− y2 = C that is x2
+ y2 =
16− C so these are circles of radius < 4, centered at the
origin.(d) Domain boundary: Circle of radius 4, centered at the
origin.
(e) Domain open, closed or neither?: Open
(f) Domain bounded or unbounded?: Bounded
-
214 CHAPTER 3. FUNCTIONS OF SEVERAL VARIABLES
7. Find the domain, range, level curves, boundary of the domain,
determineif the domain is open, closed or neither and determine if
the domain isbounded or unbounded for f (x, y) = ln
(x2 + y2
)(a) Domain: {(x, y) : x 6= 0 and y 6= 0} = R� {(0, 0)}.
(b) Range: R
(c) Level curves: Circles centered at the origin with strictly
positiveradius.
(d) Domain boundary: {(0, 0)}
(e) Domain open, closed or neither?: Open
(f) Domain bounded or unbounded?: Unbounded
8. For each function below, sketch its graph, find and sketch
its level curves.
(a) f (x, y) = x2 + 2y2
Level curves: x2
c2 +y2
c2
2
= 1 for any constant c which are ellipses.
(b) f (x, y) = x2 − y2
Level curves: x2
c2 −y2
c2 = 1 for any constant c which are hyperbolas.
(c) f (x, y) = sin(x2 + y2
)Level curves: x2 + y2 = sin−1 c for any constants c which are
circlesof radius
√sin−1 c.
(d) f (x, y) =1
x2 + y2
Level curves: x2 + y2 = 1c2 for any constant c which are circles
ofradius 1c .
9. Find an equation of the level curve of f (x, y) = 16− x2 − y2
through thepoint
(2√
2,√
2).
x2 + y2 = 10
10. Find an equation of the level curve of f (x, y) =∫ yx
dt1+t2 through the point(
−√
2,√
2).
tan−1 y − tan−1 x = 2 tan−1√
2
11. Find and sketch a typical level surface for f (x, y, z) = x+
z.They are x+ z = c. These are planes. Below is the graph when c =
4.
-
3.1. DEFINITIONS AND EXAMPLES OF FUNCTIONS OF TWOORMORE
VARIABLES215
44
22
0
2
00-2
xy
z -2-4
-2
-4
-4
4
12. Find and sketch a typical level surface for f (x, y, z) = z
− x2 − y2They are z− x2 − y2 = C or z = C + x2 + y2 that is
paraboloid along thez-axis, translated C units up. The graph below
corresponds to C = 2.
-2-4
z2
5
4
3
1
0
4
24
2
-2
-4
0
xy0
-
Bibliography
[1] Joel Hass, Maurice D. Weir, and George B. Thomas, University
calculus:Early transcendentals, Pearson Addison-Wesley, 2012.
[2] James Stewart, Calculus, Cengage Learning, 2011.
[3] Michael Sullivan and Kathleen Miranda, Calculus: Early
transcendentals,Macmillan Higher Education, 2014.
609
I Lecture NotesFunctions of Several VariablesDefinitions and
Examples of Functions of two or More VariablesClosed and Bounded
SetsGraphs of Functions of two VariablesDefining Functions of two
Variables in MapleThings to knowProblemsAnswers
Bibliography