Math 314 Lecture #11 §14.1: Functions of Several Variables A function of two variables is a rule f that assigns to each ordered pair of real numbers (x, y) in a set D a unique number f(x, y). The domain of f is the set D if specified, and is otherwise the set D of points (x, y) for which the rule f makes sense. The range of f is the set of real numbers that f realizes, i.e., {f(x, y):(x, y) ∈ D}. We often write z = f(x, y), and we call x and y the independent variables and z the dependent variable. Outcome A: Find and sketch (by hand) the domain of a function of two or more variables. √ Example. Let z = f(x, y) = x + y. Since no domain was specified, we seek for the set of points (x, y) for which the function makes sense. The square root requires that x + y ≥ 0; the domain is D = {(x, y): x + y ≥ 0}. Here is a rendering of this domain (the purple or shaded region) that includes the line x + y = 0. √ Example. Let z = f(x, y) = y − x ln(y + x). The square root requires that y−x ≥ 0 and the natural logarithm requires that y+x > 0. The domain of f is that part of the xy-plane that satisfies both of these inequalities: that is D = {(x, y): y − x ≥ 0 and y + x > 0}. Here is a rendering of this domain (the purple or shaded part) that includes part of the line y = x but excludes the line y = −x. Outcome B: Find the range of a function of two or more variables. Example. Let z = f(x, y) = p25 − x2 − y2. The domain of this is D = {(x, y): x2 + y2 ≤ 25}, i.e., all of the points on or inside the circle of radius 5 and center at the origin. The smallest value f realizes is 0; this occurs when x2 + y2 = 25. √ The largest value f realizes is 25 = 5; this occurs when x2 + y2 = 0. The function realizes every value between 0 and 5 for appropriate choices of (x, y); for 0 ≤ z ≤ 5 we choose (x, y) in D such that z2 = 25 − x2 − y2, i.e., x2 + y2 = 25 − z2. The range of f is the closed interval [0, 5]. 1 Example. Let w = f(x, y, z) = . p4 − x2 − y2 − z2 The domain is D = {(x, y, z) : 4 − x2 − y2 − z2 > 0}, i.e., all of the points inside the sphere of radius 2 with center at the origin, but not the points on this sphere. Since we can choose points (x, y, z) in D for which x2 + y2 + z2 is close to 4, there are arbitrarily large values in the range of f. √ The smallest value that f realizes is f(0, 0, 0) = 1/ 4 = 1/2. The function f realizes any value bigger than 1/2 for choices of (x, y, z) ∈ D. The range of f is the interval [1/2, ∞). Outcome C: Sketch (by hand) the graph of a function of two variables. The graph of a function z = f(x, y) of two variables with domain D is the set of points (x, y, z) in R3 such that z = f(x, y) with (x, y) ∈ D. For a simple enough function, its graph might be a plane, a cylinder, or more generally, a quadric surface. Example. Let z = f(x, y) = 3 − x2 − y2. The domain is the whole of the xy-plane, and the range is the interval (−∞, 3]. We recognize the equation z = 3 − x2 − y2 as a quadric surface. It is the equation for a circular paraboloid that opens downward with its peak at the point (0, 0, 3). Outcome D: Identify the level curves and sketch (by hand) the contour map of a function of two variables. When the graph of a function is not a quadric surface, then we need to extract additional information from the function to sketch its graph. The level curves of a function f of two variables are the curves with equations f(x, y) = k lying in the domain of f, where k is a constant in the range of f. The level curves are just the horizontal traces of the graph of f. Example. The function z = f(x, y) = x3 − y has as its domain D the whole xy-plane and as its range the whole real line. The level curves of f are the curves x3 − y = k lying in the xy-plane for any value of k. These level curves are the graphs of the cubic function y = x3 − k. Here is a sampling of these level curves for k = −3, −2, −1, 0, 1, 2, 3. Which of these curves corresponds to larger values of k? smaller values of k? Here is the graph of this function of two variables. Example. The domain of z = f(x, y) = xy2 − x3 is the whole xy-plane and the range is the whole real line. The level curves of f are the curves xy2 − x3 = k, i.e., y = ±p(k + x3)/x when x 6= 0. Here is a sampling of these level curves. Which curves corresponding to larger values of k? to smaller values of k? Here is the graph of this function. This graph is called a “monkey saddle” as it provides places for the monkey’s legs and tail. Outcome E: Describe the level surfaces of a function of three variables. The graph of a function w = f(x, y, z) of three variables lies in 4-dimensional space, and so we will not attempt to render its graph. The level surfaces of w = f(x, y, z) are the surfaces determined by the equation f(x, y, z) = k for values of k in the range of f. Example. The range of the function w = f(x, y, z) = x2 + 3y2 + 5z2 is the set of nonnegative real numbers. The level surfaces of f are the ellipsoids x2 + 3y2 + 5z2 = k for k > 0, and the single point (0, 0, 0) when k = 0. If we think of w = k ≥ 0 as “time,” then we can imagine the graph of f as an animation of its levels surfaces for increasing value of k: it starts with the point (0, 0, 0), then becomes an ever increasing ellipsoid..