Topic 3-1: Introduction to Multivariate Functions (Functions of Two or More Variables) Big Ideas. The Story So Far: Notes. Notes.

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1 Topic 3-1: Introduction to Multivariate Functions (Functions of Two or More Variables) Textbook: Section 14.1 Big Ideas The input for a multivariate function is a point. The set of all inputs is the domain of the function. The output of a multivariate function is a scalar value. The set of all outputs is the range of the function. Graphs, traces, and level curves can be used to produce visual representations of a bivariate function z = f (x, y). The Story So Far: Functions studies in Calc I and Calc II are primarily scalar-valued functions of a single variable: Input = a scalar. Output = a scalar. In the last topic, we studied vector valued functions parameterizing curves: Input = scalar. Output = a vector.

2 Motivation: Most Real-World Phenomena Involve More Than One Variable. Pythagorean Theorem: a 2 + b 2 = c 2 Ideal Gas Law: PV = nrt Newton s Universal Law of Gravitation: F = G m 1m 2 r 2 Example: Connection Between Evaluating a Function and Its Graph f (x, y) = x 2 + y 2 Plot the points ( x, y, x 2 + y 2) for the following values of x and y: (x, y) = (0, 0) (x, y) = (1, 1) (x, y) = (1, 1) (x, y) = (1, 2) Then, graph the function using an app. Try This at Home: Find an App and Graph the Following: z = 2x + 3y plane h(x, y) = xy f (x, y) = x 2 + y 2 g(x, y) = x 2 + y 2 x 2 + y 2 + z 2 = 1 saddle paraboloid cone sphere Note: The first four are graphs of functions. The last a sphere is not the graph of a function! It fails the vertical line test.

3 Some Places to Find Graphing Apps Two online apps: WolframAlpha: Surface/Level Curve Grapher: multicalc/contours/combo.html If you have a mac, look for the app grapher under utilities. Check the Resources page on the course website for additional online and downloadable graphing apps. Domain and Range of Multivariable Functions Domain: The set of all allowable inputs. Two input variables z = f (x, y), the domain is a subset of R 2 : D { (x, y) R 2} Three variables w = f (x, y, z), the domain is a subset of R 3 : D { (x, y, z) R 3} Observe: The number of input variables = the dimension of the domain. Range: The set of all possible outputs. A subset of R: R {z R} or R {w R} Traces Suppose z = f (x, y) is a function in two variables. A trace is a curve on the graph of the function obtained by holding one of the variables either x, y, or z constant. Traces are curves of intersection of the graph of the function and planes parallel to coordinate planes. Traces give a way to visualize a surface without looking at the entire graph.

4 Vertical Traces of z = f (x, y) A vertical trace is a curve on the graph of z = f (x, y) where either x or y is held constant (x = c or y = c). Vertical traces are curves of intersection of the graph of z = f (x, y) and planes x = c (planes parallel to the yz-coordinate plane) or y = c (planes parallel to the xz-coordinate plane). Example: Vertical Traces: z = x 2 + y 2 Sketch the vertical traces of z = x 2 + y 2 for x = 0, 1, 2. Sketch the vertical traces of z = x 2 + y 2 for y = 0, 1, 2. Horizontal Traces & Level Curves of z = f (x, y) A horizontal trace is a curve on the graph of z = f (x, y) at a constant height z = c. Horizontal traces are curves of intersection of the graph of z = f (x, y) and planes z = c where c is a constant (these are planes parallel to the xy-coordinate plane). A level curve of f is the projection of a horizontal trace into the xy-plane. A level curve is a curve in the domain of a function, along which the function value remains constant. Depending on context, level curves may also be called equipotentials or isotherms.

5 Contour Maps A contour map is a collection of level curves f (x, y) = c, where the difference between c-values on adjacent curves is constant. The difference between c-values on adjacent curves of a contour map is called the contour interval. The closer together contour lines (level curves) are, the steeper the graph. An example of a contour map is a topographic map. The distance between topo-lines (contour lines) represents a (constant) change in elevation. Try This at Home: Graphing Level Curves/Contour Maps Use a graphing app (or online applet) to graph level curves and/or contour maps of the following: z = 2x + 3y h(x, y) = xy f (x, y) = x 2 + y 2 g(x, y) = x 2 + y 2 z = 1 (x 2 + y 2 ) plane saddle paraboloid cone hemisphere Two online apps: WolframAlpha: Surface/Level Curve Grapher: multicalc/contours/combo.html Level Sets The notion of level curves can be extended to functions of more than one variable. For example, if: w = f (x, y, z), then the level surfaces of f are the surfaces: for c a constant. f (x, y, z) = c