Functions of Several Variables

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 was a rule which assigned a unique value to each input value. It is going to be similar for two or more variables. The only difference is that the input is not a value anymore, it is several values. Here is a more formal definition. Definition 169 Let D = {(x, y) x R and y R} be a subset of R 2. 1. A real-valued function f of two variables 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). We can extend this definition to as many variables as we wish. Example 170 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 + 107

108 CHAPTER 3. FUNCTIONS OF SEVERAL VARIABLES 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 171 You will notice that the domain is not a set of values. Rather, it is a set of pairs. sin x cos y Example 172 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} Example 173 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. 3.1.1 Closed and Bounded Sets In this section, we extend to two and higher dimensions the notion of closed interval. 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 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 174 (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 175 (interior point) We give two definitions of an interior point. 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.

3.1. DEFINITIONS AND EXAMPLES OF FUNCTIONS OF TWO OR MORE VARIABLES109 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 176 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. Example 177 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 178 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 179 (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 180 (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. Example 181 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. Definition 182 (bounded set) Let D be a subset of R 2. bounded if it is contained within some disk of finite radius. Intuitively, this means a bounded set has finite extent. D is said to be Example 183 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. Example 184 The set { (x, y) R 2 2 x 2 } is not bounded. 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.

110 CHAPTER 3. FUNCTIONS OF SEVERAL VARIABLES Figure 3.1: Plane 2x + 3y z = 2 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 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 185 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 at 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 will also look at a family of 3-D surfaces called quadric surfaces. 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 186 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 Thus, we have z = 2x + 3y + 2 z = f (x, y) = 2x + 3y + 2 The graph of this function is shown on figure 3.1.

3.1. DEFINITIONS AND EXAMPLES OF FUNCTIONS OF TWO OR MORE VARIABLES111 When we studied planes, we saw that finding the intersection of a plane with the coordinate planes was useful to help us visualize the plane. 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 187 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. Definition 188 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 3.2 shows a topographic map of Kennesaw Mountain, one can clearly see the level curves indicating where the mountain is. Figure 3.3 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 189 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. 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 ( x 2 + y 2)

112 CHAPTER 3. FUNCTIONS OF SEVERAL VARIABLES Figure 3.2: Topographic map of Kennesaw Mountain Figure 3.3: 3-D map of Kennesaw Mountain

3.1. DEFINITIONS AND EXAMPLES OF FUNCTIONS OF TWO OR MORE VARIABLES113 We would use the following syntax: f:=(x,y)->sin(x^2+y^2); 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). 3.1.4 Assignment Do odd # 1-9, all # 13-18, 29, 31, 35, 39 at the end of 11.1 in your book.