14.1 Functions of two or more variables A familiar example of a function of two variables is the area A of a rectangle, equal to the product xy of the base x and height y. We write A (x,y) = xy or A = f (x,y), where f(x,y) = xy A function of n variables is a function f that assigns a real number f(x1,.., xn) to each n-tuple (x1,.., xn) in a domain in R^n Graphing functions of two variables The graph of a function F of two variables consists of all points (a, b, f(a,b)) in R^3 for (a,b) in the domain D Of f. Assuming that F is continuous, the graph is a surface whose neigh above or below the xy-plane at (a,b) is the value of the function f(a,b) Traces One way of analyzing the graph of a function f(x,y) is to freeze the x-coordinate by setting x = a and examine the resulting curve given by z = f(a,y). Similarly, we may set y = b and consider the curve z = f(x,b) Vertical trace in the plane x=a: intersection of the graph with the vertical plane x = a, consisting of all points (a, y, f(a,y)) Vertical trace in the plane y=b: intersection of the graph with the vertical plane y = b, consisting of all points (x, b, f(x,b)) Level curves and contour maps Horizontal trace at height C: intersection of the graph with the horizontal plane z= C, consisting of the points (x, y, f(x,y)) such that f(x,y) = c Thus, the level curve corresponding to c consists of all points (x,y) in the domain Of f in the xy-plane where the function takes the value C. Each level curve is the projection onto the xy-plane of the horizontal trace on the graph that lies above it A contour map is a plot in the domain in the xy-plane that shows the level curves f(x,y) = c for equally spaced values of C. The interval m between the values of C is called the contour interval If the level curves are close together, then a small more from one level curve to the next in the xy-plane leads to a large change in height. In other words, the level curves are close together if the graph is steep Example: f (x; y) = 12 ¡ 2x ¡ 3y z = f (x; y) z = 12 ¡ 2x ¡ 3y z + 2x + 3y = 12 z = 0 ! z = f (0; 0) = 12 12 ¡ 2x ¡ 3 (0) = 0 x=6 12 ¡ 3y = 0 y=4 R Ii