Functions of more than one variable

Chapter 3 Functions of more than one variable 3.1 Functions of two variables and their graphs 3.1.1 Definition A function of two variables has two ingredients: a domain and a rule. The domain of the function is a collection of points in the x-plane. For each point (x, ) from the domain of the function, the rule should tell us how to find the function value f(x, ). Just as with functions of one variable, the rule that gives us the function value is often specified b some formula, e.g. f(x, ) =x +. The domain of a function is the set of points at which we define the function. This can in principle be an set of points in the plane. Tpicall the domain will be a rectangle, or a disc, or it could be the entire x-plane, possibl with some points and lines removed. Figure 3.1: The graph of some function, and its domain (a rectangle in this example). 3.1.2 Graphs B definition, the graph of a function z = f(x, ) is the collection of all points (x,, z) in three dimensional space that satisf the equation z = f(x, ). The graph is usuall a surface that floats above (or below) the domain of the function (see Figure 3.2). 3.1.3 Level sets The graph of a function of two variables is a surface sitting in three dimensional space, which can be difficult to draw or visualize. Instead of looking at the graph we can also consider its level sets. If c is an real number, then, b definition, the level set at level c of the function is the set of all points (x, ) in the plane that satisf f(x, ) =c. 43

44 CHAPTER 3. FUNCTIONS OF MORE THAN ONE VARIABLE Figure 3.2: The graph of some function (top), and a construction of one of its level sets (bottom). Note that b definition the level set ( at level c ) is the curve in the x-plane under the graph: it is obtained b intersecting the graph of the function with a horizontal plane at height c, and then projecting this curve of intersection onto the x-plane. Since the level set is the set of all solutions to the equation f(x, ) =c, one often uses the notation f 1 (c) ( f-inverse of c ) for the level set. We can summarize the definition in an equation: f 1 (c) = { (x, ) :f(x, ) =c }. Note that the definition sas that f 1 (c) is not a number, but a set of points! 3.1.4 Vector notation If x is the position vector of the point (x, ) in the plane, i.e. if x =( x ), then one sometimes writes f(x, ) =f( x). Phsicists have a preference for r instead of x (because the call the position vector the radius vector ), and will write f(x, ) = f( r). 3.2 Linear functions The simplest function of one variable are those of the form f(x) =ax + b. Their graphs are lines, and we called them linear functions. Definition 3.1. A linear function of two variables is a function f of the form where a, b, c are constants. z = f(x, ) =ax + b + c, (3.2.1) The graph of a linear function is alwas a plane. Indeed, the graph consists of all points (x,, z) that satisf the equation ax b + z = c, which we can write as where n x = n p, n = a b, and p = 0 0. 1 c

3.3. QUADRATIC FORMS 45 3.3 Quadratic forms Figure 3.3: The graph of a linear function z = ax + b + c. After learning about linear functions in pre-calculus one usuall goes on to quadratic functions. We will do the same for functions of two variables and stud Quadratic Forms. Just as in the one variable case where quadratic functions can have a maximum or minimum, quadratic forms provide examples of functions of two variables that can have a maximum or a minimum, or, it turns out, a third kind of min-max or saddle shape. The provide the basic profile of what we will run into when we look for local minima and maxima of functions of two variables. In particular, the technique of classifing quadratic forms b completing the square, which we will see in this section, is the ke to the second derivative test for functions of more than one variable. Definition 3.2. The general quadratic form in two variables is f(x, ) =Ax 2 + Bx + C 2, (3.3.1) where A, B, and C are constants. Depending on the values of these constants the graphs of the functions can have a number of different shapes. In addition to these quadratic forms one can also consider the more general class of quadratic functions, f(x, ) =Ax 2 + Bx + C 2 + Dx + E + F, which also have terms of degree 1 and 0. We will restrict ourselves to quadratic forms (for now). 3.3.1 Classifing quadratic forms the general procedure All quadratic forms have graphs that look like one of the examples shown above but how can we tell which it is? In other words, if Q(x, ) is a given quadratic form how can we tell if it is definite, indefinite, or semidefinite? How do we know for which (x, ) the form Q(x, ) is positive or negative? It turns out that we can alwas find out b using the trick of completing the square. The general procedure for a given quadratic form Q(x, ) =Ax 2 + Bx + C 2 is as follows: 1. If A =0, then we reall have Q = Bx + C 2 and we can factor Q as Q(x, ) =(Bx + C). 2. Assume A 0. We factor out A, and complete the square for the first two terms: Q(x, ) =A {x 2 + B A x + C A 2} { (x B = A + 2A ) 2 ( B 2A ) 2 C + A 2} { (x B = A + 2A ) 2 4AC } B2 + }{{} 4A 2 2. }{{} u 2 ±v 2 3. If 4AC B 2 > 0, then the expression in braces is positive, and we can write Q(x, ) =A(u 2 + v 2 ), where u = x + B 4AC B 2A, and v = 2. 2A Depending on the sign of A our function is alwas positive or alwas negative, and we sa the form is positive definite or negative definite.

46 CHAPTER 3. FUNCTIONS OF MORE THAN ONE VARIABLE 4. If 4AC B 2 < 0, then we have Q(x, ) =A(u 2 v 2 ), where u = x + B 2A, and v = B2 4AC. 2A When this happens we can factor the quadratic form, i.e. we have The form is indefinite. Q(x, ) =A(u + v)(u v). 5. in the onl remaining case we have 4AC B 2 =0, so that ( Q(x, ) =A x + B ) 2. 2A In this case the form is a perfect square (times A). The form is semi-definite. To understand this procedure it is perhaps best to look at how it works in some examples. 3.3.2 Classifing quadratic forms two examples An indefinite quadratic form Consider the form Q(x, ) = 3x 2 +9x +6 2. We rewrite this as follows: Q = 3x 2 +6x +9 2 = 3 ( x 2 2x 3 2) = 3 [ x 2 2x + 2 }{{} 42] complete the square = 3 [ (x ) 2 4 2] in this case we get the difference of two squares, so use a 2 b 2 =(a b)(a + b) = 3(x 2)(x +2) = 3(x 3)(x + ). This shows that Q(x, ) > 0 when > 1 3 x or < x, and Q(x, ) < 0 when x << 1 3 x. Figure 3.4: The signs of the quadratic form in example 3.3.2. A positive definite quadratic form To see a different example, consider the quadratic form Q(x, ) =2x 2 4x +6 2. B completing the square we can write it as Q(x, ) =2 { x 2 2x +3 2} =2 { x 2 2x + 2 +2 2} the square is complete =2 { (x ) 2 +2 2} =2(x ) 2 +4 2. We see that this particular quadratic form is positive definite.

3.4. FUNCTIONS IN POLAR COORDINATES R, θ 47 z = r = x 2 + 2 z z z =Φ(r) =r r x 3.4 Functions in polar coordinates r, θ Figure 3.5: Radiall smmetric functions. The graph of z = r. Recall that instead of using Cartesian coordinates (x, ) to specif the location points in the plane, we can also use polar coordinates. In man cases it is much easier to describe a function using polar coordinates than in Cartesian coordinates. To go back and forth between Cartesian and Polar Coordinates we can use the following relations The equation for θ is onl valid for x>0, where π 2 relating θ to (x, ). <θ< π 2. In other regions of the plane there are other expressions The simplest kinds of functions one can consider in polar coordinates are those that onl depend on one of those coordinates, i.e. functions that onl depend on the radius r, and functions that onl depend on the polar angle θ. Let s look at some examples of such functions. 3.4.1 Radiall smmetric functions The functions f(x, ) =x 2 + 2, g(x, ) = x 2 + 2, h(x, ) =ln ( x 2 + 2), all can be expressed in terms of the radius r onl. Namel, using r 2 = x 2 + 2,wehave f(x, ) =r 2, g(x, ) =r, h(x, ) =lnr 2 (= 2 ln r). In general, a function z = f(x, ) that can be written in terms of the radius r onl, i.e. a function for which there is some function Φ of one variable with f(x, ) =Φ(r), i.e. f(x, ) =Φ ( x2 + 2), is called a radiall smmetric function. Since a radiall smmetric function onl depends on the radius r, its level sets consist of circles centered at the origin (one exception: the origin, r =0can also be a level set, and this is obviousl not a circle but a point.) As an example, we consider the function g(x, ) = x 2 + 2 = r in more detail. The function Φ of one variable here is Φ(r) =r. We can tr to visualize the graph of g b first looking at the positive x-axis onl. There we have f(x, 0) = x 2 = x. We get the graph of g b revolving the graph of z = x around the z-axis. See Figure 3.5. 3.4.2 Functions of θ onl Here are two functions that happen to depend on the polar angle θ onl: f(x, ) =sinθ, h(x, ) =θ. We can rewrite these functions in terms of x and b using the relations between Cartesian and Polar coordinates. We get f(x, ) =sinθ = r = x2 + 2

48 CHAPTER 3. FUNCTIONS OF MORE THAN ONE VARIABLE for f, and h(x, ) =θ = arctan x for h, at least in the right half plane where x>0. A function that onl depends on θ is constant on ras emanating from the origin because the polar angle θ is constant on such ras. The level sets of such a function therefore consist of half-lines ( ras ) starting at the origin. Its graph consists of spokes attached to the z-axis. Each spoke lies above a ra in the x-plane with some polar angle θ, and is attached to the z-axis at a height given b the function value. As we var θ, the spoke rotates around the vertical axis and moves up or down, as dictated b the function. Figure 3.4.2 shows what happens for f(x, ) =sinθ. The graph of a function of θ onl consists of horizontal spokes attached to the z-axis. The graph of z =sinθ (the x-axis is coming right at us.) The function z = θ has a simpler formula in polar coordinates but actuall has a more complicated graph. Let us tr to visualize its graph: the spokes that make up the graph are horizontal, attached to the z-axis, and are at height θ. If we increase the angle θ the spokes go up at a stead rate in a wa that should remind us of a helix (see 2.5 and Figure 2.5). Based on this description its graph should look like the surface drawn in Figure 3.6. The surface is called the helicoid, and it is not the graph of a function (it fails the vertical line test. ) We could have known this from the beginning, because when we described our function as f(x, ) =θ, we should have immediatel asked which θ? The polar angle θ of an given point is onl determined up to a multiple of 2π. The graph that we have drawn of the function z = θ reflects this. To make h(x, ) =θ into an honest function we have to sa which of the man possible angles θ we choose when we are given a point. One possible choice is to alwas require the polar angle θ to lie between 0 and 2π (radians). More precisel, we can insist on 0 θ<2π. If we do this then there is a unique angle θ for each point (x, ) in the plane. The graph of this function is shown on the right in Figure 3.6. 3.5 Problems 1. Find a formula for the distance to the origin of the graph of (3.2.1). 2. Classif the following quadratic forms as definite, indefinite, or other, b completing the square. Determine the zero set for each of these quadratic forms. (a) f(x, ) =x 2 +2 2 (b) Q(x, ) =x 2 2 (c) g(x, ) =x 2 4x +3 2 (d) Q(s, t) =9s 2 36st +81t 2 (e) M(α, β) = 1 2 α2 αβ + β 2. (f) Q(x, ) =x + 2 (g) Q(x, ) =x 2 +2x (h) For which values of the constant k is the quadratic form Q(x, ) =x 2 +2kx + 2 positive definite? 3. Which functions of two variables z = f(x, ) are defined b the following formulae? Find draw the domain of each function (the largest domain on which the definition would make sense). Tr to sketch their graphs. Draw the level sets for each function. (a) z = x (b) z x 2 =0

3.5. PROBLEMS 49 x x Figure 3.6: The graph of z = θ is the helicoid. It is not the graph of a function, but one can extract a function b choosing a branch of the function. One possible choice, drawn here on the right, is to restrict the polar angle θ to the interval 0 θ<2π. There are man other possible choices. (c) z 2 x =0 (d) z x 2 2 =0 (e) z 2 x 2 2 =0 (f) xz =1 (g) x/z 2 =1 (h) x + + z 2 =0 (i) x + + z 2 =1 4. The following expressions are all equal to the polar angle θ in some region of the x-plane. Explain wh the expression gives θ, and identif in which region this holds. (a) θ = arctan x (b) θ = π + arctan x (c) θ =2π + arctan x (d) θ = π 2 arctan x (e) θ = arcsin. x2 + 2 5. Describe and explain the relation between the graph of the function = g(x) of one variable, and the corresponding function f(x, ) =g ( x 2 + 2) of two variables. What do the level sets of f(x, ) look like? For instance, if g(x) =x, then f(x, ) = x 2 + 2 : what is the relation between the graphs of g and f? 6. Find the largest domain on which the following functions of two (or occasionall three) variables can be defined: (a) f(x, ) = 9 x 2 + 2 4 (b) f(x, ) =arcsin(x 2 + 2 2) (c) f(x, ) = x (d) f(x, ) = x (e) f(x,, z) =1/ xz (f) f(x, ) = 16 x 2 4 2 7. Here are two sets of level curves with levels z = 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4. One is for a function whose graph is a cone (z = x 2 + 2 ), the other is for a paraboloid (z = x 2 + 2 ). Which is which? Explain. 8. If = g(x) is an function of one variable, then a function of the form f(x, t) =g(x ct) is often called a traveling wave with wave speed c and profile g. Let g be an non constant function of our choice and describe the movie presented b the function f(x, t) =g(x ct) (can t choose? Then tr Agnesi s witch g(x) = 1 1+x 2.) The number c is called the wave speed. If c>0 is the motion to the left or to the right? Explain. 9. Let Q be the square in the plane consisting of all points (x, ) with x 1, 1. This problem is about the so-called distance function to Q. This function is defined

50 CHAPTER 3. FUNCTIONS OF MORE THAN ONE VARIABLE as follows: f(x, ) is the distance from the point (x, ) to the point in Q nearest to (x, ). Which point in Q is nearest to (0, 1 2 )? Which is closest to (0, 2)? Which is closest to (3, 4)? Compute f(0, 1 2 ), f(0, 2) and f(3, 4)). What is the zero set of f? Draw the level sets of f at levels 1, 1,2,and3. Describe the general level set f(x, ) =c where c is an arbitrar number.