BSU Math 275 Notes Topic 3.1: Introduction to Multivariate Functions (Functions of Two or More Variables) Textbook Section: 14.1 From the Toolbox (what you need from previous classes): Know the meaning of domain and range for scalar valued functions of a single variable, y = f (x) (the type of function studied in Calc I/II). Be able to evaluate and graph functions of a single variable. Learning Objectives (New Skills) & Important Concepts Learning Objectives (New Skills): Evaluate scalar valued functions of two and three variables. Find the domain and range of scalar valued functions of two and three variables. Given a graph of a function z = f (x, y), be able to sketch vertical and horizontal traces. Use a graphing app to graph functions and level curves/contour maps of functions of two variables. Be able to find equations for and sketch the level curves of a function z = f (x, y), and to construct a contour map. Be able to match the graph of a function z = f (x, y) with its contour map. For a function z = f (x, y), use a contour map to determine the value of a function, the change in a function s value between two points in the domain, and the average rate of change of a function s value with respect to distance in the function s domain.
Important Concepts: The input for a multivariate function is a point. The domain of a multivariate function is the set of all allowable inputs. The dimension of the domain is the same as the number input variables. So, the domain of a function z = f (x, y) is a subset of R 2 (2-dimensional), and the domain of a function w = f (x, y, z) is a subset of R 3 (3- dimensional). The output of a multivariate function is a scalar value. The range of a multivariate function is set of all possible outputs, which is a subset of R. The graph of a function of two variables z = f (x, y) is the set of all points (x, y, f (x, y)) where (x, y) is a possible input (in the domain of f ). The graph z = f (x, y) is a surface (two-dimensional object) in R 3. For a function z = f (x, y) of two variables, traces are curves in the graph of the function that are generated by holding one of the variables x, y, or z constant. Vertical traces of a function z = f (x, y) are generated by holding either x or y constant, producing curves on the graph that are parallel to the xz- or yz-coordinate planes. These are the curves of intersection of the graph z = f (x, y) with the vertical planes x = c or y = c (c is a constant). Horizontal traces of a function z = f (x, y) are generated by holding z constant, producing curves on the graph that are parallel to the xy-coordinate plane. These are the curves of intersection of the graph z = f (x, y) with the horizontal planes z = c (c is a constant). Level curves of a function z = f (x, y) are curves in the xy-plane defined by the equations c = f (x, y). They are the projections of horizontal traces into the domain of the function. A collection of level curves with a constant change in the value c between adjacent curves is called a contour map or contour diagram. The difference of the c-values between adjacent curves is called the contour interval. 2
The Big Picture Our interest in multivariate functions (functions with more than one independent variable) arises from the fact that most real-world processes depend on more than one parameter. There are both similarities and differences between scalar valued functions of a single variable y = f (x), and scalar valued functions of more than one variable. For example, both single- and multivariable functions have a domain and a range; and for both, the range is a subset of the real numbers R. However, the domain of a function depends on the number of independent variables. The domain of a function of a single variable is a subset of R. The domain of a function of two variables is R 2. The domain of a function of three variables is R 3. In general, the domain of a function of n variables is R n. Another difference lies in graphical representations of multivariate functions. A function of two variables z = f (x, y) has a graph that is a two-dimensional surface in R 3. It also can be represented by traces, and by level curves and contour diagrams. The graphs of functions of three or more variables are not useful in visualizing the function. Functions of three variables have a limited visual representation in traces and level sets, but these are not nearly as complete as for functions of two variables. More Details There are several ways of visually representing a function of two variable z = f (x, y): The graph of a function of two variables z = f (x, y) is a twodimensional surface in R 3. We can sketch these by hand, or use a computer to graph them. The graph of a function of three variables w = f (x, y, z), however, is a three-dimensional hyper-surface in R 4. Since we cannot visualize objects in four dimensions, these do not give us useful visual information. The same is true for functions of more than three variables: their graphs are n-dimensional hypersurfaces in R n+1, and do not provide useful visual information. A trace is a curve generated in the graph of a function by holding all but one of the independent variables constant. We will use traces 3
in the future when we define partial derivatives, and again when we define the area element of a parameterized surface. For functions of two variables z = f (x, y), traces are generated by intersecting the graph of the function with planes parallel to the coordinate planes. Vertical traces of a function z = f (x, y) are the curves of intersection of the graph of the function and planes parallel to the xz- or y z-coordinate planes. These planes have equations y = c (parallel to the xz-plane) or x = c (parallel to the yz plane), so the vertical traces are defined by the equations z = f (x, c), y = c or z = f (c, y), x = c. (These are the traces we will use when defining partial derivatives.) Horizontal traces of a function z = f (x, y) are the curves of intersection of the graph of the function and planes parallel to the xy-coordinate plane. These planes have equations z = c, so the horizontal traces are defined by the equations c = f (x, y), z = c. A level set is a subset of a function s domain over which the function s value remains constant. For functions of two variables z = f (x, y), these are curves (called level curves) in the xy-plane, defined by the equations f (x, y) = c, z = 0, or simply f (x, y) = c. They are projections of the horizontal traces into the xy-plane. The value of a function over a level curve is constant. If the function z = f (x, y) represents the height of a surface relative to the xy-plane, then an ant walking along a level curve would look up (or down) at the surface, and the distance from the ant to the surface would not change. If the function z = f (x, y) represents the temperature of a flat plate at the location (x, y), then an ant walking along a level curve would experience constant temperature (in this case of temperature functions, level curves are called isotherms ). Contour maps (or, contour diagrams) are collections of level curves, where the change in the function value on adjacent level curves (called the contour interval) is constant. This allows you to read the steepness of the graph of the function, by seeing how closely spaced curves are: the closer the spacing, the steeper the graph. Contour maps can be used to determine the change in a function s value between two points in the domain, and the average rate of 4
change of a function between two points in the domain. Real-world examples of contour maps are topographic maps used to indicate terrain and maps of air pressure used in meteorology. There are only limited ways of visualizing functions of three variables. Level sets of a function of three variables are surfaces in R 3, which can be graphed. Coloring these level sets to correspond to function s values can help organize the information. However, level sets of a function of three variables does not provide the same level of visual clarity that a contour map does for a function of two variables. The reason it becomes so difficult to visually represent functions of three or more independent variables is due to the fact that the total number of dimensions in the domain plus the dimension needed for the range is at least four; but we cannot visualize dimensions higher than three. 5