functions of several variables -...

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

)


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.


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.


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


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.


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);


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}?


(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.


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


(x, y) ∈ R2 : 1 < x2 + y2 < 4}?

There are two boundaries: the circle x2 +y2 = 1 and the circle x2 +y2 = 4.


(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


(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



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



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


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


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.


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.

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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

functions of several variables -...

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