Functions of Several Variables

Transcription

1 Chapter 9 Functions of Several Variables Functions that depend on several input variables first appeared in the S-I-R model at the beginning of the course. Usuall, the number of variables has not been an issue for us. For instance, when we introduced the derivative in chapter 3, we used partial derivatives to treat functions of several variables in a parallel fashion. However, when there are questions of visualization and geometric understanding, the number of variables does matter. Ever variable adds a dimension to the problem one wa or another. For eample, if a function has two input variables instead of one, we will see that its graph is a surface rather than a curve. This chapter deals with the geometr of functions of two or more variables. We start with graphs and level sets. These are the basic tools for visualization. Then we turn to microscopic views, and see what form the microscope equation takes. Finall, we consider optimization problems using both direct visual methods and dnamical sstems. The problem of visualizing a function of several variables 9. Graphs and Level Sets The graph at the right comes from a model that describes how the average dail temperature at one place varies over the course of a ear. It shows the temperature A in F, and the time t in months from Januar. As we would epect, the temperature is a periodic function (which we can write as A(t)), and its period is months. Furthermore, temperature A months t 5 Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

2 5 CHAPTER 9. FUNCTIONS OF SEVERAL VARIABLES Underground temperatures fluctuate less the lowest temperature occurs in Februar (when t or 3) and the highest in Jul (when t 7 or 9). This is about what we would epect. However, all these temperature fluctuations disappear a few feet underground. Below a depth of 6 or 8 feet, the temperature of the soil remains about 55 F ear-round! Between ground level and that depth, the temperature still fluctuates, but the range from low to high decreases with the depth. Here is what happens at some specific depths. A 7 temperature 6 5 d = 6 feet d = feet d = foot The phase shifts with the depth Graphing temperature as a function of depth months d = feet Notice how the time at which the temperature peaks gets later and later as we go farther and farther underground. For eample, at d = feet the highest temperature occurs in September (t 9), not Jul. It literall takes time for the heat to sink in. In chapter 7 we called this a phase shift. The lowest temperature shifts in just the same wa. At a depth of feet, it is colder in March than in Januar. Thus A is reall a function of two variables, the depth d as well as the time t. To reflect this addition, let s change our notation for the function to A(t, d). In the figure above, d plas the role of a parameter: it has a fied value for each graph. We can reverse these roles and make t the parameter. This is done in the figure on the top of the net page. It shows us how the temperature varies with the depth at fied times of ear. Notice that, in April and October, the etreme temperature is not found on the surface. In October, for eample, the soil is warmest at a depth of about 9 inches. The lower figure on the page is a single graph that combines all the information in these two sets of graphs. Each point on the bottom of the bo of the bo corresponds to a particular depth and a particular time. The height of the surface above that point tells us the temperature at that depth and time. For eample, suppose ou want to find the temperature feet t Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

3 9.. GRAPHS AND LEVEL SETS 53 A 7 t = 6 (Jul) temperature 6 5 t = 9 (October) t = 3 (April) 3 t = (Januar) 6 8 depth d below surface at the beginning of Jul. Working from the bottom front corner of the bo, move feet to the right and then 6 months toward the back. This is the point (d, s) = (, 6). The height of the graph above this point is the temperature A that we want. 3 6 months Reading a surface graph 7 6 temperature 5 A 53 3 (d, s) = (, 6) depth 6 8 Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

4 Grid lines are slices of the surface that show how the function depends on each variable separatel temperature Comparing the surface to its slices 5 CHAPTER 9. FUNCTIONS OF SEVERAL VARIABLES There is a definite connection between this surface and the two collections of curves. Imagine that the bo containing the surface graph is a loaf of bread. If ou slice the loaf parallel to the left or right side, this slice is taken at a fied depth. The cut face of the slice will look like one of the graphs on page 5. These show how the temperature depends on the time at fied depths. If ou slice the loaf the other wa parallel to the front or back face then the time is fied. The cut face will look like one of the graphs on page 53. The show how the temperature depends on the depth at fied times. The grid lines on the surface are precisel these slice marks. Here is the same graph seen from a different viewpoint. Now time is measured from the left, while depth is measured from the back. The temperature is still the height, though. One advantage of this view is that it shows more clearl how the peak temperature is phase-shifted with the depth. We now have two was to visualize how the average dail temperature depth 6 depends on the time of ear and 8 the depth below ground. One is the surface graph itself, and the other is a collection of curves that are slices of months the surface. The surface gives us an overall view, but it is not so eas to read the surface graph to determine the temperature at a specific time and depth. Check this ourself: what is the temperature feet underground at the beginning of April? The slices are much more helpful here. You should be able to read from either collection of slices that A F = = Eamples of Graphs The purpose of this section is to get some eperience constructing and interpreting surface graphs. To work in a contet, look first at the functions = and =. The provide us with standard eamples of a minimum and a maimum when there is just one input variable. Let s consider now the corresponding eamples for two input variables. Besides an ordinar maimum and an ordinar minimum, we will find a third tpe called Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

5 9.. GRAPHS AND LEVEL SETS 55 a minima that is completel new. It arises because a function can have a minimum with respect to one of its input variables and a maimum with respect to the other. A minimum: z = + At the origin (, ) = (, ), z =. At an other point, either or is non-zero. Its square is positive, so z >. Consequentl, z has a minimum at the origin. The graph of this function is a parabolic bowl whose lowest point sits on the origin. As alwas, the grid lines are slices, made b fiing the value of or. For eample, if = c, then the slice is z = +c. This is an ordinar parabolic curve z A maimum: z = For an and, the value of z in this eample is the opposite of its value in the previous one. Thus, z is everwhere negative, ecept at the origin, where its value is. Thus z has a maimum at the origin. Its graph is an upside-down bowl, or peak, whose highest point reaches up and touches the origin. Grid lines are the curves z = c and z = c. These are parabolic curves that open downward z A minima: z = Suppose we fi at =. This slice has the equation z =, so it is an ordinar parabola (that opens upward). Thus, as far as the input is concerned, z has a minimum at the origin. Suppose, instead, that we fi at =. Then we get a slice whose equation is z =. It is also an ordinar parabola, but this one opens downward. As far as is concerned, z has a maimum at the origin. It is clear from the graph how upward-opening slices in the -direction fit together with downward-opening slices in the -direction. Because of the shape of the surface, a minima is commonl called a saddle, or a saddle point. z Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

6 56 CHAPTER 9. FUNCTIONS OF SEVERAL VARIABLES Here are two slices of z = shown in more detail. Points in the bo have three coordinates: (,, z). If we set = we are selecting the points = z z = The points where = c form a vertical plane parallel to the, z-plane The points where = c form a vertical plane parallel to the, z-plane of the form (,, z). These make up the, z-plane. On this plane the equation z = becomes - simpl z =. The graph of this equation is a - curve in the, z-plane specificall, the parabola - shown. The situation is similar if is given some other fied value. For eample, = specifies the points (,, z). These describe the plane that forms the front face of the bo. The equation z = becomes z = 6. The curve tracing out the intersection of the saddle with the front of the bo is precisel the graph of z = 6. If = we get the points (,, z) that make up the, z-plane. On this plane the equation simplifies to z =, and its graph is the parabolic curve shown. Giving a different fied value leads to similar results. A good eample is =. The points (,, z) lie on the plane that forms the left side of the bo. The equation becomes z = 6 there, and this is the parabolic curve marking the intersection of the saddle with the left side of the bo. As ou can see, it is valuable for ou to be able to generate surface graphs ourself. There are now a number of computer utilities which will do the job. Some can even rotate the surface while ou watch, or give ou a stereo view. However, even without one of these powerful utilities, ou should tr to generate the slicing curves that make up the grid lines of the surface. Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

7 9.. GRAPHS AND LEVEL SETS 57 A cubic: z = 3 Slices of this graph are downward-opening parabolas (when = c) and are cubic curves that have the same shape (when = c). Notice that each cubic curve has a maimum and a minimum, and each parabola has z = = =.8 = z = = ± = ± = ±.5 a maimum. The surface graph itself has a peak where the cubics have their maimum, but it has a saddle where the cubics have a minimum. Do ou see wh? The saddle point is a minima for z = 3 : z has a minimum there as a function of alone but a maimum as a function of alone. The surface has a saddle 5 z The small figures on the right show the same surface as the large figure; the just show it from different viewpoints. As a practical matter, ou should look at these surfaces the wa ou would look at sculpture: walk around them b generating diverse views. - See the graph from different viewpoints Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

8 58 CHAPTER 9. FUNCTIONS OF SEVERAL VARIABLES Energ of the pendulum: E = cosθ + v 3 E v - - π θ π π 3π E 3 v - - π π θ π 3π This function first came up in chapter 7, where it was used to demonstrate that a dnamical sstem describing the motion of a frictionless pendulum had periodic solutions. It was used again in chapter 8 to clarif the phase portrait of that dnamical sstem. The function E varies periodicall with θ, and ou can see this in the graph. The minimum at the origin is repeated at (θ, v) = (π, ), and so on. The graph also has a saddle at the point (θ, v) = ( π, ). This too repeats with period π in the θ direction. The figure at the left is the same surface with part cut awa b a slice of the form v = c. These slices are sine curves: E = cosθ + c. Slices of the form θ = c are upward-opening parabolas. From this viewpoint, the saddle points show up clearl. One wa to describe what happens to a real pendulum that is, one governed b frictional forces as well as gravit is to sa that its energ runs down over time. Now, at an moment the pendulum s energ is a point on this graph. As the energ runs down, that point must work its wa down the graph. Ultimatel, it must reach the bottom of the graph the minimum energ point at the origin (θ, v) = (, ). This is the stable equilibrium point. The pendulum hangs straight down (θ = ) and is motionless (v = ). The graph gives us an abstract but still vivid and concrete wa of thinking of the dissipation of energ. Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

9 9.. GRAPHS AND LEVEL SETS 59 From Graphs to Levels There is still another wa to picture a function of two variables. To see how it works we can start with an ordinar graph. On the right is the graph of z = f(, ) = 3, the cubic function we considered on page 57. This graph looks different, though. The difference is that points are shaded according to their height. Points at the bottom are lightest, points at the top are darkest. Notice that the flat, -plane is shaded eactl like the graph above it. For instance, the dark spot centered at the point (, ) = (, ) is directl under the peak on the graph. The other dark patch, near the right edge of the plane, is under the highest visible part of the surface. Consequentl, the shading on the, -plane gives us the same information as the graph. In other words, the intensit of shading at (, ) is proportional to the value of the function f(, ). The figure in the, -plane is called a densit plot. Think of the intensit of shading as the densit of ink on the page. Here are densit plots of the standard minimum, maimum, and minima. Compare these with the z Densit plots z = + z = z = graphs on page 55. The third densit plot is the most interesting. From the center of the, -plane, the shading increases to the right and left. Therefore, Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

10 5 CHAPTER 9. FUNCTIONS OF SEVERAL VARIABLES A sample plot z has a minimum in the horizontal direction. However, the shading decreases above and below the center. Therefore, z has a maimum in the vertical direction. Thus, ou reall can see there is a minima at the origin. Tr our hand at reading the densit plot on the left below. You should see two maima (directl above and below the origin), a minimum (at the origin itself), and two saddles (to the right and the left of the origin). The function defining the plot is f(, ) = ( + ( ) 3)(3 ( + ) ) Can ou visualize what the graph looks like? This densit plot should help ou, and ou can also construct slices b setting = c and = c. The slices = and = are especiall useful. With them ou could determine the eact coordinates of the maima and the saddles. These densit plots show a checkerboard pattern because Mathematica (the computer program that produces them) shades each little square according to the value of the function at the center of the square. This pattern is an artefact; it is not inherent to a densit plot. From densities to contours In a densit plot, the shading varies smoothl with the value of the function. This is accurate, but it ma be a bit difficult to read. On the right ou see a modified densit plot. There is still shading, but there are now just a few distinct shades. This makes a sharp boundar between one shade and the net. The boundar is called a contour, or a level. The figure itself is called a contour plot. The two maima on the vertical line = stand Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

11 9.. GRAPHS AND LEVEL SETS 5 out more clearl on the contour plot. Also, the contour lines around the two saddles help us see that the function has a minimum in the vertical direction and a maimum in the horizontal direction. Once we have contour lines to separate one densit level from the net, we can even dispense with the shading. The figure on the right is just the contour plot from the opposite page, minus the shading. The contour lines, or level curves, now stand out clearl. On each contour, the value of the function is constant. This is also called a contour plot. There is some loss of information here, however. For eample, we can t tell where the value of the function is large and where it is small. Nevertheless, the nested ovals on the vertical line = do tell us that there is either a maimum or a minimum at the center of each nest For reference purposes, here are the contour plots for the standard minimum, maimum, and saddle. In the first two cases, the contours are concentric ovals. These look the same, so onl one is illustrated. The other two pictures show a saddle. In general, the contours around a saddle are a famil of hperbolas. However, it is possible for one of the contour lines to pass eactl through the minima point. That contour is a pair of crossed lines, as shown in the version on the right. You should compare these contour plots with the densit plots of the same functions on page 59, and with their graphs on page 55. Contours of the standard functions two functions but one plot z = + z = two plots of a single function z = Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

12 5 CHAPTER 9. FUNCTIONS OF SEVERAL VARIABLES Contours are horizontal slices of a graph There is a direct connection between the contour plot of a function and its graph. Contours are horizontal slices of the graph, just as grid lines are vertical slices. Below, we use the standard functions z = and z = + to illustrate the connection. Notice that ever contour down in the, -plane lies eactl below, and has the same shape as, a horizontal slice of the graph. This picture eplains wh contours are called level curves Energ of the pendulum, again To get some more eperience with contour plots, we return to the energ function of the pendulum: E(θ, v) = cosθ + v. From the contour plot alone ou should be able to see that E has either a minimum or a maimum at (θ, v) = (, ), and another at (π, ). The contours also provide evidence that there is a saddle (minima) near (θ, v) = ( π, ) and (π, ). It is also apparent that E is a periodic function of θ. v - - π π π 3π θ Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

13 9.. GRAPHS AND LEVEL SETS 53 What ou should find most striking about this plot, however, is the wa it resembles the phase portrait of the pendulum (chapter 8, pages 7 7). Ever level curve here looks like a trajector of the dnamical sstem. This is no accident. We know from chapter 8 that the energ is a first integral for the dnamics. In other words, energ is constant along each trajector this is the law of conservation of energ. But each level curve shows where the energ function has some fied value. Therefore, each trajector must lie on a single energ level. We can carr the connection between contours and trajectories even further. Closed trajectories correspond to oscillations of the pendulum. But the closed trajectories are the closed contours, and these are the ones that surround the minimum. In particular, the are low energ levels. B contrast, at higher energies (E >, in fact), the pendulum will just continue to spin in what ever direction it was moving initiall. Thus, each high energ level is occupied b two trajectories one for clockwise spinning and one for counter-clockwise. Technical Summar medium energ oscillation low energ oscillation v high energ counter-clockwise spin high energ clockwise spin Energ contours are trajectories of the dnamics θ The eamples we have seen so far were meant to introduce some of the common was of visualizing a function z = f(, ). To use them most effectivel, though, ou need to know more precisel how each is defined. We review here the definition of a graph, a densit plot, a contour plot, and a terraced densit plot. Graph. The graph of z = f(, ) lies in the 3-dimensional space with coordinates (,, z). To construct it, take an input (, ). Identif this with the point (,, ) in the, -plane (which is defined b the condition z = ). The corresponding point on the graph lies at the height z = f(, ) above the, -plane. This point has coordinates (,, f(, )). The graph is the set of all points of the form (,, f(, )). This is a -dimensional surface. z (,, f(, )) (,, ) Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

14 5 CHAPTER 9. FUNCTIONS OF SEVERAL VARIABLES Densit plot. The densit plot of z = f(, ) lies in the -dimensional, -plane. Choose an rectangle where the function is defined, and let m and M be the minimum and maimum values, respectivel, of f(, ) on the rectangle. Define ρ(, ) = f(, ) m M m. Then ρ satisfies ρ(, ) on the rectangle; it is called a densit function (ρ is the Greek letter rho). In the densit plot, the densit of ink or darkness at (, ) is ρ(, ). Contour plot. A contour of z = f(, ) is the set of points in the, -plane where f has some fied value: f(, ) = c. f(, ) = c That fied value c is called the level of the contour. (The two solid ovals in the figure at the left make up a single contour.) A contour is also called a level curve. A contour plot of f is a collection of curves f(, ) = c j in the, -plane. In the plot it is customar to use constants c, c,... that are equall spaced; that is, the interval between one c j and the net alwas has the same value c. Terraced densit plot. This is a contour plot in which the region between two adjacent contours is shaded with ink of a single densit. If the contours are at levels c and c, then the densit that is tpicall chosen is the one for the level half-wa between these two that is, for their average (c +c )/. Each region is called a terrace. Often, a terraced densit plot is drawn in color, using different colors for each terrace. Television weather programs use terraced densit plots to describe the temperature forecast for a large region. We find densit plots everwhere. A photograph is a densit plot of the light that fell on the film when it was eposed. A newspaper half-tone illustration is also a densit plot of an image. Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

15 9.. GRAPHS AND LEVEL SETS 55 The pros and cons Each of these modes of visualization has advantages and disadvantages. All are reasonabl good at indicating the etremes (the maima and minima) of a function. A contour plot needs some additional information for eample, a label on each contour to indicate its level to distinguish between maima and minima. However, if ou want to know the numerical value of f(, ) at a particular point (, ), a contour plot with labels offers more precision than a densit plot. It s usuall better than a graph, too. Overall, a graph has the biggest visual impact, but there is a cost. It takes three dimensions to represent the graph of a function of two variables, but onl two to represent a plot. The cost is that etra dimension. It means that we cannot draw the graph of a function of three variables. That would take four mutuall perpendicular aes an impossibilit in our three-dimensional space. However, we can produce a contour plot. Contours of a Function of Three Variables We pause here for a brief glimpse of a large subject. B analog with the definition for a function of two variables, we sa that a contour of the function f(,, z) is the set of points (,, z) that satisf the equation Plots are visuall economical in comparison to graphs Contours and levels f(,, z) = c, for some fied number c. We call c the level of the contour. Let s find the contours of w = + + z. This is completel analogous The standard minimum to the function + with two inputs. (What do the contours of + look like?) In particular, w has a minimum when (,, z) = (,, ). As the following diagram shows, w = + + z is the square of the distance from the origin to (,, z). (We use the Pthagorean theorem twice: once for p and once for q.) Consequentl, all points (,, z) where w has a fied value z ais ais q p (,, z) p = +, z q = p + z = + + z ais = w. Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

16 56 CHAPTER 9. FUNCTIONS OF SEVERAL VARIABLES lie a fied distance from the origin. Specificall, w = + + z = c is the following set: c > : the sphere of radius c entered at the origin; c = : the origin itself; c < : the empt set. The contours are spheres The contour plot of w = + + z is thus a nest of concentric spheres, as shown in the illustration below. The value of w is constant on each sphere. (The tops of the spheres have been cut awa so ou can see how the spheres nest; the whole thing resembles an onion.) Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

17 9.. GRAPHS AND LEVEL SETS 57 Below is the contour plot of another standard function with three input variables: w = f(,, z) = + z. A quarter of each surface has been cut awa so ou can see how the surfaces nest together. Note that w = is a cone, and ever surface with w < consists of two disconnected (but congruent) pieces an upper half and a lower half. You should compare this function to the standard minima in two variables. The three-variable function w has a minimum with respect to both of the variables and, while it has a maimum with respect to z. (Do ou see wh? The arguments are eactl the same as the were for two input variables on page 55.) Furthermore, the contours of are a famil of hperbolas, and the contours of + z are surfaces obtained b rotating these hperbolas about a common ais. A standard minima The contours are hperbolic shapes z w = upper half w = 9 w = w = w = lower half Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

18 58 CHAPTER 9. FUNCTIONS OF SEVERAL VARIABLES When there are three input variables, the contours are surfaces It is a general fact and our two eamples provide good evidence for it that a single contour of a function of three variables is a surface. Thus a contour is a curve or a surface, depending on the number of input variables. We often use the term level set (rather than a level curve or a level surface) as a generic name for a contour. Eercises In man of these eercises it will be essential to have a computer program to make graphs, terraced densit plots, and contour plots of functions of two variables.. a) Use a computer to obtain a graph of the function z = sin sin on the domain π, π. How man maimum points do ou see? How man minimum points? How man saddles? b) Determine, as well as ou can from the graph, the location of the maimum, minimum, and saddle points.. Continuation. Make the domain π π, π π and answer the same questions ou did in the previous eercise. (Does the graph look like an egg carton?) 3. Obtain a terraced densit plot (or a contour plot) of z = sin sin on the domain π π, π π. Locate the maimum, minimum, and saddle points of the function. Do these results agree with those from the previous eercise?. Obtain the graph of z = sin cos on the domain π, π. How does this graph differ from the one in eercise? In what was is it similar? 5. Obtain the graph of z = + when,. Locate all the minimum, maimum, and saddle points in this domain. [Note: the minimum is on the boundar!] 6. Continuation. Obtain a terraced densit plot (or contour plot) for the function in the previous eercise, using the same domain. Use the plot to locate all the minimum, maimum, and saddle points. Compare our results with those of the previous eercise. Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

19 9.. GRAPHS AND LEVEL SETS a) Obtain the graph of z = on the domain,. What is the shape of the graph? b) Graph the same function of the domain 6,. What is the shape of the graph? How does this graph compare to the one in part (a)? 8. a) Continuation. Sketch three different slices of the graph of z = in the -direction. What do the slices have in common? How are the different? b) Answer the same questions for slices in the -direction. 9. a) Obtain the graph of z = ; choose the domain ourself. Where does the graph intercept the z-ais? b) Describe the vertical slices of this graph in the -direction and in the -direction.. Describe the vertical slices of the graph of z = p + q + r in the -direction and in the -direction.. a) Compare the contours of the function z = + to those of z = +. b) What is the shape of the graph of z = +? Decide this first using onl the information ou have about the contours. Then use a computer to obtain the graph.. a) Compare the contours of the function z = to those of z =. b) What is the shape of the graph of z =? Decide this first using onl the information ou have about the contours. Then use a computer to obtain the graph. 3. a) Obtain a contour plot of the function z = + +. b) What is the shape of the graph of z = ++? Decide this first using onl the information ou have about the contours. Then use a computer to obtain the graph.. a) Obtain a contour plot of the function z = Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

20 53 CHAPTER 9. FUNCTIONS OF SEVERAL VARIABLES b) What is the shape of the graph of z = +3+? Decide this first using onl the information ou have about the contours. Then use a computer to obtain the graph. 5. a) Obtain a contour plot of the function z = + +. b) What is the shape of the graph of z = ++? Decide this first using onl the information ou have about the contours. Then use a computer to obtain the graph. 6. Complete this statement: The function f(, ; p) = + p +, which depends on the parameter p, has a minimum at the origin when and a minima when. 7. a) Obtain the graph and a terraced densit plot of the function z = What is the shape of the graph? b) What is the shape of the contours? Indicate how the contours fit on the graph. 8. a) Obtain the graph and a terraced densit plot of the function z = What is the shape of the graph? b) What is the shape of the contours? Indicate how the contours fit on the graph. 9. a) Obtain the graph and a terraced densit plot of the function z = What is the shape of the graph? b) What is the shape of the contours? Indicate how the contours fit on the graph.. Obtain the graph of z = f(, ) = on the domain 3 3, 3 3. Does this function have a maimum or a minimum or a saddle point? Where?. a) Continuation. Sketch slices of the graph of z = in the -direction, for each of the values =,,,, and. What is the general shape of each of these slices? b) Repeat part (a), but make the five slices in the -direction that is, fi instead of. Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

21 9.. GRAPHS AND LEVEL SETS 53. Continuation. Show how the slices ou obtained in the previous eercise fit (or appear) on the graph ou obtained in the eercise just before that one. 3. a) Continuation. Let = u + v and = u v. Epress z in terms of u and v, using the fact that z =. Then obtain the graph of z as a function of the new variables u and v. b) What is the shape of the graph ou just obtained? Compare it to the graph of z = ou obtained earlier.. Can ou draw a network of straight lines on the saddle surface z =? 5. Obtain a terraced densit plot of z =. How do the contours of this plot fit on the graph of z = ou obtained in a previous eercise? 6. The graphs of z = +5 + and z = 3 intersect in a curve. What is the shape of that curve? 7. The graphs of z = + 3 and z = intersect in a curve. What is the shape of that curve? 8. The graphs of z = and z = intersect in a curve. What is the shape of that curve? First integrals 9. A hard spring described b the dnamical sstem d dt = v, dv dt = c β3, has a first integral of the form E(, v) = c + β + v. This is the energ of the spring. (See chapter 7.3, especiall eercise 3, page 5.) a) Let c = 6 and β =. Obtain the graph of E(, v) on a domain that has the origin at its center. Locate all the minimum, maimum, and saddle points in this domain. Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

22 53 CHAPTER 9. FUNCTIONS OF SEVERAL VARIABLES b) What is the state of the spring (that is, its position and its velocit v) when it has minimum energ? 3. A soft spring described b the dnamical sstem d dt = v, dv dt = 5 +, has an energ integral of the form (See eercise 6, page 55.) E(, v) = 5 ln( + ) + v. a) Obtain the graph of E(, v). Eperiment with different possibilities for the domain until ou get a good representation. b) Obtain a terraced densit plot of E(, v) over the same domain ou chose in part (a). Compare the two representations of E. c) Does the spring have a state of minimum energ? If so, where is it? d) Does the spring have a state of maimum energ? Eplain our answer. 3. a) The Lotka Volterra equations. According to eercise 33 of chapter 7.3 (page 58), the function E(, ) =. ln +. ln.5. is a first integral of the dnamical sstem =..5, =... Obtain the graph of E on the domain 5, 5. (Wh not enlarge the domain to 5, 5?) b) Find all maimum, minimum, and saddle points on this graph. What is the connection between the maimum of E and the equilibrium point of the dnamical sstem? c) Obtain a contour plot of E on the same domain as in part (a). Compare the contours of E and the trajectories of the dnamical sstem. (This reveals a conservation of energ for the solutions of the Lotka-Volterra equations.) Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

23 9.. GRAPHS AND LEVEL SETS a) Continuation. Here is another first integral of the same dnamical sstem as in the previous eercise: H(, ) =.. ep(.5 +. ). Obtain the graph of H and compare it to the graph of E in the previous eercise. b) Obtain a contour plot of H, and compare the contours to the trajectories of the dnamical sstem. Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

24 53 CHAPTER 9. FUNCTIONS OF SEVERAL VARIABLES 9. Local Linearit Local linearit is the central idea of chapter 3: it sas that a graph looks straight when viewed under a microscope. Using this observation we were able to give a precise meaning to the rate of change of a function and, as a consequence, to see wh Euler s method produces solutions to differential equations. At the time we concentrated on functions with a single input variable. In this section we eplore local linearit for functions with two or more input variables. Magnifing a graph Microscopic Views Consider the cubic f(, ) = 3 that we used as an eample in the previous section. We ll eamine both the graph and the plot of f under a microscope. In the figure below we see successive magnifications of the graph near the point where (, ) = (.5, ). The initial graph, in the left rear, is drawn over the square.5.5,.5.5. With each magnification, the portion of the surface we see bends less and less. The graph approaches the shape of a flat plane Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

25 9.. LOCAL LINEARITY 535 Contour plots for f(, ) = 3 appear below. Again, we magnif near the point (, ) = (.5, ). Each window below is a small part of the window to its left. In the large scale plot, which is the first one on the left, the contours are quite variable in their direction and spacing. With each magnification, that variabilit decreases. The contours become straight, parallel, and equall spaced The process of magnification thus leads us to functions whose graphs are flat and whose contours are straight, parallel, and equall-spaced. As we shall now see, these are the linear functions. Linear Functions A linear function is defined b the wa its output responds to changes in the input. Specificall, in chapter we said = f() is linear if = m. This is the simplest possibilit: changes in output are strictl proportional to changes in input. The multiplier m is both the rate at which changes with respect to and the slope of the graph of f. Eactl same idea defines a linear function of two or more variables: the change in output is strictl proportional to the change in an one of the inputs. Responses to changes in input The definition Definition. The function z = f(,,..., n ) is linear if there are multipliers p, p,..., p n for which z = p, z = p,..., z = p n n. Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

26 536 CHAPTER 9. FUNCTIONS OF SEVERAL VARIABLES Partial and total changes There is one multiplier for each input variable. The multipliers are constants and the are, in general, all different. The definition describes how z responds to each input separatel. We call each p j j a partial change. The multiplier p j = z/ j is the corresponding partial rate of change. Of course, several input variables ma change simultaneousl. In that case, the total change in z will just be the sum of the individual changes produced b the several variables: z = p + p + + p n n. Another wa to describe a linear function Of course, if the total change of a function satisfies this condition, then each partial change has the form p j j. (If onl j changes, then all the other k must be. So z becomes simpl p j j.) Consequentl, the function must be linear. In other words, we can use the formula for the total change as another wa to define a linear function. Alternate definition. The function z = f(,,..., n ) is linear if there are multipliers p, p,...,p n for which z = p + p + + p n n. Formulas for linear functions From the definition to a formula Given the partial rates of change and an initial point When z = f(,,..., n ) is a linear function, we know how z depends on the changes j, but that doesn t tell us eplicitl how z itself is related to the input variables j. There are several was to epress this relation as a formula, depending on the nature of the information we have about the function. For the sake of clarit, we ll develop these formulas first for a function of two variables: z = f(, ). The initial-value form. Suppose we know the value of a linear function at some given point called the initial point and we also know its partial rates of change. Can we construct a formula for the function? Suppose z = z when (, ) = (, ), and suppose the partial rates of change are If we let p = z and q = z. =, =, z = z z, Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

27 9.. LOCAL LINEARITY 537 then we can write z z = z = p + q = p ( ) + q ( ). This is the initial-value form of a linear function. For eample, if the initial point is (, ) = (, 3), z = 5, and the partial rates of change are z/ =, z/ = +, the equation of the linear function can be written z 5 = ( ) + ( 3). The intercept form. This is a special case of the initial-value form, in which the initial point is the origin: (, ) = (, ), z = r. The formula becomes z r = p + q, or z = p + q + r. As we shall see, the graph of this function in,, z-space passes through the point (,, z) = (,, r) on the z-ais. This point is called the z-intercept of the graph. Sometimes we simpl call the number r itself the z-intercept. Notice that, with a little algebra, we can convert the previous eample to the form z = + +. This is the intercept form, and the z-intercept is z =. If there are n input variables,,,..., n, instead of two, and an initial point has coordiantes,,..., n, then a linear equation has the following forms: initial-value: z z = p ( ) + p ( ) + + p n ( n n), z-intercept: z = p + p + + p n n + r. Given the partial rates of change and the z-intercept The form of a linear function of n variables The graph of a linear function On the left at the top of the net page is the graph of the linear function z = + +. The graph is a flat plane. In particular, grid lines parallel to the -ais (which represent vertical slices with = c) are all straight lines with the same slope z/ =. The other grid lines (with = c) are all straight lines with the same slope z/ = +. Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

28 538 CHAPTER 9. FUNCTIONS OF SEVERAL VARIABLES 8 6 z slope = q 3 3 z = + + r slope = p z = p + q + r The definition of a linear function implies that its graph is a flat plane On the right, above, is the graph of the general linear function written in intercept form: z = p+q+r. The graph is the plane that can be identified b three distinguishing features: it has slope p in the -direction; it has slope q in the -direction. it intercepts the z-ais at z = r; Let s see how we can deduce that the graph must be this plane. First of all, the partial rate z/ tells us how z changes when is held fied. But if we fi = c, we get a vertical slice of the graph in the -direction. The slope if that vertical slice is z/ = p. Since p is constant, the slice is a straight line. The value of = c determines which slice we are looking at. Since z/ doesn t depend on, all the slices in the -direction have the same slope. Similarl, all the slices in the -direction are straight lines with the same slope q. The onl surface that can be covered b a grid of straight lines in this wa is a flat plane. Finall, since z = r when (, ) = (, ), the graph intercepts the z-ais at z = r. Contours of a linear function Each contour is a straight line A contour of an function f(, ) is the set of points in the, -plane where f(, ) = c, for some given constant c. If f = p + q + r, then a contour has the equation p + q + r = c or = p q + c r. q Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

29 9.. LOCAL LINEARITY 539 This is an ordinar straight line in the, -plane. Its slope is p/q and its -intercept is (c r)/q. (If q = we can t do these divisions. However, this causes no problem; ou should check that the contour is just the vertical line = (c r)/p.) To construct a contour plot, we must give the constant c a sequence of equall-spaced values c j, with c j+ = c j + c. This generates a sequence of straight lines p + q + r = c j, or = p q + c j r. q = c j r q c q p + q + r = cj p + q + r = cj + p + q + r = cj + These lines all have the same slope p/q, so the are parallel. (Notice the value of c doesn t affect the slope.) The -intercept of the j-th contour is (c j r)/q. Therefore, the distance along the -ais between one intercept and the net is c/q. The contours are thus straight, parallel, and equall-spaced. (You should check that this is still true if q =.) Note that the figure at the left, above, is drawn with c > but q <. Geometric interpretation of the partial rates What happens to the graph or the contour plot if ou double one of the partial rates of change of a linear function? The graph on the right, below, shows the effect of doubling the partial rate with respect to of the function z = + +. As ou can see, the slope in the -direction Partial rates and partial slopes z z z = + + z = + + Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

30 5 CHAPTER 9. FUNCTIONS OF SEVERAL VARIABLES The overall tilt of a graph is altered Partial rate and the spacing of contours is doubled (from to ). Had we increased the partial rate b a factor of, the slope would have increased b a factor of as well. Notice that the slope in the -direction is not affected. Nevertheless, the overall tilt of the graph has been altered. We shall have more to sa about this feature in a moment, when we introduce the gradient of a linear function to describe the overall tilt. A change in the partial rates has a more comple effect on the contour plot. Perhaps it is more surprising, too. To make valid comparisons, we have constructed all three plots below with the same spacing between levels (namel z = ). Notice how the levels meet the - and -aes in the plot on the left (z = tfrac + + ). For each unit step we take along the -ais, the z-value increases b. This is the meaning of z/ = +. B contrast, we have to take two unit steps along the -ais to produce the same size change in z. Moreover, z decreases b when increases b. This is the meaning of z/ =. In particular, the relativel wide spacing between z-levels along the -ais reflects the relative smallness of z/. 3 z = 7 z = 6 z = 5 z = z = 3 3 z = 7 z = 6 z = 5 z = z = 3 z = z = 3 z = z = 8 z = 6 z = z = The larger the partial rate, the closer the contours z = + + z = + + z = + + Therefore, when we double the size of z/ as we do in the middle plot we should cut in half the spacing between z-levels along the -ais. As ou can see, this is eactl what happens. Notice that the spacing along the -ais is not altered. Consequentl, the contours change direction and the get packed more closel together. Suppose we double both partial rates as we do in the plot on the right. Then the spacing between contours is cut in half along both aes. Because the change is uniform, the contours keep their original direction. Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

31 9.. LOCAL LINEARITY 5 The Gradient of a Linear Function B making use of the concept of a vector, introduced in the last chapter, we can construct still another geometric interpretation of the partial rates of a linear function. This vector is called the gradient, and it is defined in the following wa. The vector of partial rates Definition. The gradient of a linear function z = f(, ) is the vector whose components are its partial rates of change: ( z grad z = z =, z ). The gradient is perhaps the most concise and useful tool for describing the growth of a function of several variables. To get an idea of the role that it plas, consider this question: In what direction should we move from a given point in the, -plane so that the value of a linear function increases most rapidl? The direction of most rapid growth Of course, the answer will depend on the linear function. Let s use z = z = and start from the point (, ) = (.,.6). We can make z undergo a ver large change simpl b moving ver far from this point. 3 Therefore, to make valid comparisons, we will restrict ourselves to motions that carr us eactl one unit of distance in various directions. The vectors in the figure at the right show some of the possibilities. Their tips lie on a circle of radius. 3 Thus, to choose the direction in which z increases most rapidl, we must simpl find the point on this circle where the value of z is largest. The contour line at this level must be tangent to the circle. The vector perpendicular to 3 this contour line (see the second figure) therefore points in the direction of most rapid growth. Since perpendiculars have negative reciprocal slopes, and since all the contour lines have slope +/, it follows that the vector must have slope /. 3 At the left is a magnified view of this vector. We know <, =, and ( ) + ( ) =. Thus =, so ( ) + ( ) =. This implies 5 ( ) =, so = 5, = 5. high z Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

32 5 CHAPTER 9. FUNCTIONS OF SEVERAL VARIABLES Thus, among all the motions (, ) we have considered, we obtain the greatest change in z b choosing ( ) (, ) =,. 5 5 The magnitude of To determine how large this change is, we can use the alternate definition of most rapid growth a linear function (see page 536) z = z z + = + = =. The gradient vector quickl gives us all this information. First of all, the 3 grad z gradient vector has the value ( z grad z =, z ) = (, ). 3 Since its slope is / =, we see that it does indeed point in the direction Information from of most rapid growth. Consequentl, it is also perpendicular to the contour the gradient line. Furthermore, its length gives the maimum growth rate. We can see this b calculating the length using the Pthagorean theorem: ( z ) ( ) z 5 5 length = + = + = =. Our findings with this eample point to the following conclusion. Theorem. The gradient of the linear function z = p + q + r is perpendicular to its contour lines. It points in the direction in which z increases most rapidl, and its length is equal to the maimum rate of increase. A proof Let s see wh this is true. According to the observation on the previous page, the direction of most rapid increase will be perpendicular to the contour lines. The gradient of z = p + q + r is the vector z = (p, q). Its slope is q/p. On page 539 we saw that the slope of the contour lines is p/q. Since these slopes are negative reciprocals, the gradient is indeed perpendicular to the contour lines. Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

33 9.. LOCAL LINEARITY 53 To determine the maimum rate of increase, we must see how much z increases when we move eactl unit of distance in the gradient direction. The gradient vector is (p, q), and its length is p + q. Therefore, the vector (, ) = ( p p + q, ) q p + q is unit long and in the same direction as the gradient. The increase in z along this vector is z = p +q = p p p + q +q q p + q = p + q p + q = p + q. This is the length of the gradient vector, so we have confirmed that the length of the gradient is equal to the maimum rate of increase. End of the proof z = ++ z = ++ z = + + Shown above are the three linear functions we ve alread eamined. In each case the gradient vector is perpendicular to the contours, and it gets longer as the space between the contours decreases. This is to be epected because the space between contours is also an indicator of the maimum rate of growth of the function. Widel-spaced contours tell us that z changes relativel little as and change; closel-spaced contours tell us that z changes a lot as and change. The connection between the gradient and the graph is particularl simple. Since the gradient (which is a vector in the, -plane) points in the direction of greatest increase, it points in the direction in which the graph is tilted up. If we project the gradient vector onto the graph, as in the figure at the top of the net page, it points directl uphill. Putting it another wa, we can Contour spacing and the length of the gradient The gradient points directl uphill Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8

34 5 CHAPTER 9. FUNCTIONS OF SEVERAL VARIABLES z Local linearit Eceptions 8 sa that the gradient shows us the overall tilt of the graph. There are two parts to this information. 6 First, the direction of the gradient tells us which wa the graph is tilted. Second, the length tells us how steep the graph is. The figure at the left combines all the visual ele- 3 ments we have introduced to analze a linear func- tion: contours, graph, and gradient. Stud it to see how the are related. 3 The Microscope Equation Local linearit Let s return to arbitrar functions of two variables that is, ones that are not necessaril linear. First we looked at magnifications of their graphs and contour plots under a microscope. We found that the graph becomes a plane, and the plot becomes a series of parallel, equall-spaced lines. Net, we saw that it is precisel the linear functions which have planar graphs and uniforml parallel contour plots. Hence this function is locall linear. Of course, not ever function is locall linear, and even a function that is locall linear at most points ma fail to be so at particular points. We have alread seen this with functions of a single variable in chapter 3. For eample, g() = /3 is locall linear everwhere ecept the origin. It has a sharp spike there. The two-variable function f(, ) = ( + ) /3 has the same sort of spike at the origin. The two graphs help make it clear that g is just a slice of f z.5.5 z z = g() = /3 z = f(,) = ( + ) /3 Copright 99, 8 Five Colleges, Inc. DVI file created at :, 3 Januar 8