Functions of Several Variables

Transcription

1 Chapter 3 Functions of Several Variables 3.1 Definitions and Examples of Functions of two or More Variables In this section, we extend the definition of a function of one variable to functions of two or more variables. You will recall that a function is a rule which assigns a unique output value to each input value. It is similar for functions of two or more 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 : R 2 R or f : R 3 R. Here is a more formal definition. Definition Let D = {(x, y) : x R and y R} be a subset of R A real-valued function of two variables f : D R is a rule which assigns to 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 is understood that the domain is the largest possible set of input values that is 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 assigns to 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 R 3. We can extend this definition to as many variables as we wish. 203

2 204 CHAPTER 3. FUNCTIONS OF SEVERAL VARIABLES Example Find the domain of f (x, y) = sin ( x 2 + y 2) x 2 + y 2 x 2 + y 2 is always defined, therefore sin ( x 2 + y 2) is always defined. Since x 2 + y 2 0 except when x = y = 0, it follows that f is always defined except at (0, 0). So, its domain is R 2 \ {(0, 0)}. Remark You will notice that the domain is not a set of values. Rather, it is a set of pairs. sin x cos y Example Find the domain of g (x, y) = x y The numerator is always defined, so is the denominator. However, the denominator cannot be zero. It is zero when y = x. The domain is the set R 2 {(x, x) : x R}. We could also write that the domain is { (x, y) R 2 : y x }. Example Find the domain of h (x, y) = x ln ( y x 2) ln is defined when its argument is positive. So, we see that for h to be defined, we must have y x 2 > 0 y > x 2 So, the domain of h is the portion of the xy-plane inside the parabola y = x 2. It is the yellow region in figure 3.1. We could mwrite that the domain is { (x, y) R 2 : y > x 2} Closed and Bounded Sets In this section, we extend to two and higher dimensions the notion of closed and open intervals. You will recall that a closed interval on the real line is an interval 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 R 2 and R 3. In this section, we will not present this material very thoroughly. This is usually done in an advanced calculus or in a real analysis class. The intent here is to give the reader an idea of what the notion of closed set in R 2 and R 3 is. Definition (boundary point) We extend the notion of the end point of an interval to higher dimensions. Such points are called boundary points. 1. Let D be a subset of R 2. A boundary point of D is a point (a, b) such that every disk centered at (a, b) contains both points of D and points not in D. 2. Let D be a subset of R 3. A boundary point of D is a point (a, b, c) such that every sphere centered at (a, b, c) contains both points of D and points not in D. Definition (interior point) We give two definitions of an interior point.

3 3.1. DEFINITIONS AND EXAMPLES OF FUNCTIONS OF TWO OR MORE VARIABLES205 Figure 3.1: Domain of h (x, y) = x ln ( y x 2)

4 206 CHAPTER 3. FUNCTIONS OF SEVERAL VARIABLES 1. Let D be a subset of R 2 or R 3. An interior point of D is a point in D which is not on the boundary of D. The set of interior points of a given set is called the interior of that set. 2. Let D be a subset of R 2. A point P of D is an interior point of D if there exists a disk containing P which is included in D. 3. Let D be a subset of R 3. A point P of D is an interior point of D if there exists a sphere containing P which is included in D. Remark Let us make the following remarks: 1. This agrees with our intuitive definition of a boundary. If you were on the boundary between two countries, stepping on one side would put you in one country, 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 of a set be in the set. We will see in the examples the boundary points of a set are not always in the set. In fact, it is a special property a set has when 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 and a sphere in space. They represent a region around a point. When we do not specify the dimension, we will use the term ball. Thus a ball can be an interval, a disk or a sphere, depending on which dimension we are in. The term ball is also used in higher dimensions. Example The boundary of the disk defined by x 2 + y 2 1 is the circle x 2 + y 2 = 1. Its interior is the disk x 2 + y 2 < 1. Example The boundary of the disk defined by x 2 + y 2 < 1 is the circle x 2 + y 2 = 1. Its interior is the disk x 2 + y 2 < 1. You will note that the above two sets are different, yet they have the same boundary. The main difference is that the first set contains its boundary, the second does not. This is an important fact to remember. A boundary point of a set does not necessarily belong to the set. Definition (closed set) We extend the notion of a closed interval to higher dimensions. Let D be a subset of R 2 or R 3. D is said to be closed if it contains all its boundary points. Definition (open set) We extend the notion of an open interval to higher dimensions. Let D be a subset of R 2 or R 3. D is said to be open if every point of D is an interior point of D.

5 3.1. DEFINITIONS AND EXAMPLES OF FUNCTIONS OF TWO OR MORE VARIABLES207 Figure 3.2: A Closed Set Example The disk defined by x 2 + y 2 1 is closed because it contains its boundary, the circle x 2 + y 2 = 1. Even if one point from the boundary were omitted, the set would no longer be closed. Example The disk defined by x 2 + y 2 < 1 is open. Every point is an interior point. Remark When a set is closed, we represent its boundary with a solid line as shown in figure 3.2. When it is open, we represent its boundary with a dashed line as shown in figure 3.3. Definition (bounded set) Let D be a subset of R 2. D is said to be bounded if it is contained within some disk of finite radius. A subset D of R 3 is said to be bounded if it is contained within some sphere of finite radius. In general, a subset D of R n is said to be bounded if it is contained within a ball of finite radius. Intuitively, this means a bounded set has finite extent. Example The disk defined by x 2 + y 2 1 is bounded, it is contained in any disk centered at the origin with radius larger than one. It ex- Example The set { (x, y) R 2 : 2 x 2 } is not bounded. tends to infinity in the y-direction. We will see that sets which are both closed and bounded have an important property related to finding extreme values later on in the chapter.

6 208 CHAPTER 3. FUNCTIONS OF SEVERAL VARIABLES Figure 3.3: An Open Set Graphs of Functions of two Variables Given a function of two variables f (x, y), for each value of (x, y) in the domain of f, we can plot the point (x, y, z) where z = f (x, y). The set of points in space we obtain is called the graph of f. Definition The graph of a function of two variables f (x, y) is the set of 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 in the sense that it helps to visualize the behavior of f. The graph of a function of two variables is a surface in space. Unfortunately, graphing a function of two variables is far more diffi cult than a function of one variable. Fortunately for us, we have technology which facilitates this task. Though we will not spend a lot of time graphing functions of two variables, we will explore some of the issues 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 0, we can solve for z and rewrite the plane as a function of two variables. Example Find the function f (x, y) so that the plane 2x+3y z +2 = 0 can be written as z = f (x, y). Sketch its graph using technology. We simply solve for z to obtain z = 2x + 3y + 2

7 3.1. DEFINITIONS AND EXAMPLES OF FUNCTIONS OF TWO OR MORE VARIABLES209 Figure 3.4: Plane 2x + 3y z = 2 Thus, we have z = f (x, y) = 2x + 3y + 2 The graph of this function is shown on figure 3.4. Remark If we cannot solve for z as we did above, we can still graph the corresponding function using an implicit graph. many graphing programs have the capability of generating implicit graphs. If the graph is fairly simple, finding its intersection with the coordinate planes 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 of the plane. 2. To find the intersection with the yz-plane, set x = 0 in the equation of the plane. 3. To find the intersection with the xz-plane, set y = 0 in the equation of the 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 line 3y z = 0 and the xz-plane in the line 2x z = 0. When a surface is more complicated to visualize, we do not limit ourselves to finding how it intersects the coordinate planes. We look how it intersects any plane parallel to one of the coordinate axes. The curve we obtain are called the traces or cross-sections of the surface. Definition The traces or cross-sections of a surface z = f (x, y) are the intersection of that surface with planes parallel to the coordinate planes, that is planes of the form x = C 1, y = C 2, z = C 3 where C 1, C 2 and C 3 are constants.

8 210 CHAPTER 3. FUNCTIONS OF SEVERAL VARIABLES Figure 3.5: Topographic map of Kennesaw Mountain Definition The curves obtained by finding the intersection of a surface z = 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 plot made of the contour curve is called a contour plot. The level curves are curves where the z value is constant. Level curves are uses for example in mapping, to indicate the altitude. The altitude is the same everywhere on a level curve. Figure 3.5 shows a topographic map of Kennesaw Mountain, one can clearly see the level curves indicating where the mountain is. Figure 3.6 shows a 3-D rendering of the same area. On weather maps, level curves represent isobars, that is areas where the atmospheric pressure is the same. Example Consider the surface z = f (x, y) = x 2 + y 2. Its level curves are of the form x 2 + y 2 = C, they are circles. It also intersects planes parallel to the xz-planes in the curves z = x 2, which is a parabola. It intersects planes parallel to the yz-plane in the curves z = y 2, which is also a parabola 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 ( x 2 + y 2) We would use the following syntax: f:=(x,y)->sin(x^2+y^2);

9 3.1. DEFINITIONS AND EXAMPLES OF FUNCTIONS OF TWO OR MORE VARIABLES211 Figure 3.6: 3-D map of Kennesaw Mountain Once defined, the user can evaluate the function for specific values of x and y simply by typing f (2, 3) for example. One can also use the function name to plot 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 the correct syntax. If z is not defined explicitly in terms of x and y. one must use implicitplot3d (see Maple help for the correct syntax) 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 Problems Make sure you have read, studied and understood what was done above before attempting the problems. 1. What is the boundary of the set { (x, y) R 2 : 1 < x 2 + y 2 < 4 }? 2. What is the boundary of the set { (x, y) R 2 : 1 x 2 + y 2 4 }? 3. Find the domain, range, level curves, boundary of the domain, determine bounded or unbounded for f (x, y) = y x.

10 212 CHAPTER 3. FUNCTIONS OF SEVERAL VARIABLES 4. Find the domain, range, level curves, boundary of the domain, determine bounded or unbounded for f (x, y) = 4x 2 + 9y Find the domain, range, level curves, boundary of the domain, determine bounded or unbounded for f (x, y) = xy. 6. Find the domain, range, level curves, boundary of the domain, determine 1 bounded or unbounded for f (x, y) =. 16 x2 y 2 7. Find the domain, range, level curves, boundary of the domain, determine bounded or unbounded for f (x, y) = ln ( x 2 + y 2). 8. For each function below, sketch its graph, find and sketch its level curves. (a) f (x, y) = x 2 + 2y 2 (b) f (x, y) = x 2 y 2 (c) f (x, y) = sin ( x 2 + y 2) (d) f (x, y) = 1 x 2 + y 2 9. Find an equation of the level curve of f (x, y) = 16 x 2 y 2 through the point ( 2 2, 2 ). 10. Find an equation of the level curve of f (x, y) = y dt x 1+t through the point ( ) 2 2, 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 x 2 y Answers 1. What is the boundary of the set { (x, y) R 2 : 1 < x 2 + y 2 < 4 }? There are two boundaries: the circle x 2 +y 2 = 1 and the circle x 2 +y 2 = What is the boundary of the set { (x, y) R 2 : 1 x 2 + y 2 4 }? There are two boundaries: the circle x 2 +y 2 = 1 and the circle x 2 +y 2 = Find the domain, range, level curves, boundary of the domain, determine bounded or unbounded for f (x, y) = y x. (a) Domain: R 2 (b) Range: R

11 3.1. DEFINITIONS AND EXAMPLES OF FUNCTIONS OF TWO OR MORE 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, determine bounded or unbounded for f (x, y) = 4x 2 + 9y 2 (a) Domain: R 2 (b) Range: [0, ) (c) Level curves: They are the curves in the xy-plane 4x 2 + 9y 2 = 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, determine bounded or unbounded for f (x, y) = xy (a) Domain: R 2 (b) Range: R (c) Level curves: They are the curves y = c x (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, determine 1 bounded or unbounded for f (x, y) = 16 x 2 y 2 (a) Domain: The set of points satisfying 16 x 2 y 2 > 0 that is x 2 +y 2 < 16 so it is the inside of the disk of radius 4, centered at the origin. (b) Range: [ 1 4, ) (c) Level curves: They are the curves 16 x 2 y 2 = C that is x 2 + y 2 = 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

12 214 CHAPTER 3. FUNCTIONS OF SEVERAL VARIABLES 7. Find the domain, range, level curves, boundary of the domain, determine bounded or unbounded for f (x, y) = ln ( x 2 + y 2) (a) Domain: {(x, y) : x 0 and y 0} = R {(0, 0)}. (b) Range: R (c) Level curves: Circles centered at the origin with strictly positive radius. (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) = x 2 + 2y 2 x Level curves: 2 c + y2 2 c 2 2 = 1 for any constant c which are ellipses. (b) f (x, y) = x 2 y 2 x Level curves: 2 c y2 2 c = 1 for any constant c which are hyperbolas. 2 (c) f (x, y) = sin ( x 2 + y 2) Level curves: x 2 + y 2 = sin 1 c for any constants c which are circles of radius sin 1 c. 1 (d) f (x, y) = x 2 + y 2 Level curves: x 2 + y 2 = 1 c 2 radius 1 c. for any constant c which are circles of 9. Find an equation of the level curve of f (x, y) = 16 x 2 y 2 through the point ( 2 2, 2 ). x 2 + y 2 = Find an equation of the level curve of f (x, y) = y dt x 1+t through the point ( ) 2 2, 2. tan 1 y tan 1 x = 2 tan 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.

13 3.1. DEFINITIONS AND EXAMPLES OF FUNCTIONS OF TWO OR MORE VARIABLES z y x Find and sketch a typical level surface for f (x, y, z) = z x 2 y 2 They are z x 2 y 2 = C or z = C + x 2 + y 2 that is paraboloid along the z-axis, translated C units up. The graph below corresponds to C = z y x 2 4 4

14 Bibliography [1] Joel Hass, Maurice D. Weir, and George B. Thomas, University calculus: Early transcendentals, Pearson Addison-Wesley, [2] James Stewart, Calculus, Cengage Learning, [3] Michael Sullivan and Kathleen Miranda, Calculus: Early transcendentals, Macmillan Higher Education,