Chapter 3: Functions of Several Variables
Section 3.1 - Functions and Their Graphs
Section 3.2 - Limits and Continuity
Section 3.3 - Quadric Surfaces
The space curve in Chapter 2 moved visualization in 3D from the line to the curve. The study of the quadric surfaces moves visualization from the plane to those surfaces defined by equations that are quadratic in at least one of x, y, and z. But before this, the notion of a function of several variables, and the resulting graphs that can be used to represent them, must be mastered.
If heights z are determined by a function of two variables so that z=fx,y, then a surface whose heights z, drawn over the xy-plane, is a representation of f. Such a surface is said to be defined explicitly. An equation of the form gx,y,z=0 will in general define, say, z=zx,y, so that g=0 defines the surface z implicitly.
The tools for graphing and analyzing surfaces, defined either explicitly or implicitly, are level curves and plane sections, curves that Maple can provide on graphs of the surfaces. For the explicitly defined surface, z=fx,y, the level curves are the curves defined implicitly by the equation z=fx,y=c, where c is constant. When these curves are projected down onto the xy-plane, the resulting graph is called a contour map. The Maple command for constructing such a map is the implicitplot command in the plots package.
A function of three independent variables might represent a physical quantity such as the temperature or air pressure in a room, and would be given by w=gx,y,z. The three independent variables x,y,z require the three axes of Cartesian 3D space, and there is no fourth axis for the dependent variable w. Hence, visualizing such a function is problematic. One solution is to graph a level surface defined by g=c, where c is constant. Maple does not have a command for drawing a set of level surfaces - the implicitplot3d command in the plots package is not the generalization of the implicitplot command, merely a different implementation of the same functionality.
Having introduced the idea of a multivariate function, the typical course defines the bivariate limit, the extension of the concept of limit to the plane. The definition for limx,y→a,bfx,y=L requires the values of the function f to be close to the number L whenever the point x,y is close enough to the point a,b. This definition does not amount to iteration, that is, to taking the limit in x separate from the limit in y. The limit must be path independent, holding as neighborhoods of x,y shrink down to a,b. Fortunately, Maple has functionality for computing the bivariate limit of at least rational functions.
And of course, once there is a definition for the limit for functions of several variables, the notion of continuity follows just as it did for functions of a single variable.
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