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Chapter 3: Functions of Several Variables
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Section 3.1: Functions and Their Graphs
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Essentials


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In singlevariable calculus, the notation $y\=f\left(x\right)$ is used to indicate that the values taken on by $y$ are determined by the function whose name is $f$ and whose rule is the expression $f\left(x\right)$. Such a function is said to be defined explicitly. Alternatively, the equation $yf\left(x\right)\=0$, or more generally $g\left(x\,y\right)\=0$, usually defines a function $y\=y\left(x\right)$ implicitly.

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The typical multivariate calculus course deals with functions having either two or three independent variables. The equation of a plane, namely, $axplus;byplus;czplus;dequals;0$, provided the previous chapters with an example of a linear function of two variables. The function $z\=f\left(x\,y\right)\=\left(axplus;byplus;d\right)sol;c$ is defined explicitly, whereas the equation $axplus;byplus;czplus;dequals;0$ defines this function implicitly.

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The domain of discourse for the remainder of this Guide is delineated in Table 3.1.1.

Description

Number of Independent Variables

Two

Three

Explicit

$z\=f\left(x\,y\right)$

$w\=f\left(x\,y\,z\right)$

Implicit

$g\left(x\,y\,z\right)\=0$

$g\left(x\,y\,z\,w\right)\=0$

Parametric

$x\=x\left(s\,t\right)\,y\=y\left(s\,t\right)\,z\=z\left(s\,t\right)$

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Table 3.1.1 Domain of discourse for functions of several variables



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In Table 3.1.1, the implicit cases $g\=0$ define an explicit function locally, under suitable conditions. The equation $g\left(x\,y\,z\right)\=0$ can generally be solved for one of the three variables in terms of the other two; the equation $g\left(x\,y\,z\,w\right)\=0$, one of the four variables in terms of the other three.

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The explicitly defined $z\=f\left(x\,y\right)$ can also be interpreted as parametrically defined, with the parameters $s$ and $t$ taken as $x$ and $y$, respectively. The parametric representation is then $x\=x\,y\=y\,zequals;z\left(xcomma;y\right)$.

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In singlevariable calculus, graphs of $f\left(x\right)$ are used to study the properties of the function $y\=f\left(x\right)$. Similarly, functions of two and three independent variables yield to graphical analysis. Table 3.1.2 lists the most common graphs that will appear in the multivariate calculus course.${}$

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•

An explicit function of two variables such as $f\left(x\,y\right)$ is graphed by the plot3d command as a surface, where the height on the surface corresponds to the value of $f$ at the point $\left(x\,y\right)$. On this surface, lines of equal height are called contours, or level curves.

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The plot3d and contourplot3d commands draw the contours in place on the surface. Of course, these 3D graphs can be rotated so that when viewed from above, it appears that the contours are all in the plane $z\=0$. The image so generated is called a contour plot or a contour map. The contourplot command draws a 2dimensional image of the contours where all the level curves have been projected to the plane $z\=0$. The plot3d command exists at toplevel, but the contourplot and contourplot3d commands are in the plots package.

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The intersection of a surface with planes of the form $x\=c$ or $y\=c$, $c$ constant, are called plane (or cross) sections, and are best drawn by the CrossSection command in the Student MultivariateCalculus package. The
tutor, also a part of the Student MultivariateCalculus package, implements the CrossSection command, which draws both level curves and plane sections.

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An implicitly defined function of two variables, one determined by an equation of the form $g\left(x\,y\,z\right)\=0$, is graphed as a surface in Maple with the implicitplot3d command, also a part of the plots package.

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A function of three independent variables needs four axes for its graph, so is a challenge for the human imagination which seems restricted to seeing the world with just three spatial coordinates. (The philosopher Kant used the phrase categorical imperative to describe this limitation.) As difficult as it is to imagine such a graph, functions of three independent variables are abundant in the physical sciences. At each point in a room whose geographical coordinates are $\left(x\,y\,z\right)$ there is a temperature $T$, a pressure $p$, a density $\mathrm{\ρ}$, etc. Each of these physical quantities thereby generates a function of the form $w\=f\left(x\,y\,z\right)$, where $w$ is the physical quantity.

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One way to visualize the function of three variables is by graphing its level surfaces. Connect all the points in the room that have the same temperature ${T}_{0}$ to form the isothermal surface $f\left(x\,y\,z\right)\={T}_{0}$, or ${T}_{0}f\left(x\,y\,z\right)\=g\left(x\,y\,z\right)\=0$. Thus, each level surface of a function of three variables can be viewed as an implicitly defined function of two variables. According to Kant, it's the best that can be done.

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Unfortunately, Maple does not have a command that draws a collection of level surfaces for a function of four variables. The closest Maple comes to providing that functionality is the implicitplot3d command, which draws a single surface defined implicitly by an equation of the form $g\left(x\,y\,z\right)\=0$. Undoubtedly, the reason for this is that the command was created before Maple could render a surface with transparency. If the level surfaces are one inside the other, like the layers of an onion, without transparency only the outer surface is visible!

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The Plot Builder generally can be used interactively to produce the requisite graphs needed in the study of functions of one and several variables. The various options for the underlying commands can be accessed through the various panels of the Plot Builder, but there are occasions where direct access to specifically the "right" set of options is easier to achieve by a direct application of the relevant command.

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Representing the value of a function on a particular contour in a contour map is a challenge. Although it would be ideal if the numeric value of each contour could appear on the graph, this would make for a very crowded display. All the mechanisms in Table 3.1.2 at best use an indeterminate shading to indicate how function values vary across contours. The
task template is a more direct and interactive tool for this purpose. This, and the other graphing tools in Table 3.1.2 are illustrated in the examples below.

The generalization of the idea of level curves and surfaces to higher dimensions, is that of the level set. This Guide, however, sticks with the two more specific terms: level curves, and level surfaces.


Examples


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Example 3.1.1

Graph the surface determined by $z\=f\left(x\,y\right)\=103{x}^{2}7{y}^{2}$.

Example 3.1.2

Obtain a contour map of the function $z\=f\left(x\,y\right)\=103{x}^{2}7{y}^{2}$.

Example 3.1.3

Graph the surface determined by $z\=f\left(x\,y\right)\=103{x}^{2}7{y}^{2}$, $z\in \left[10\,10\right]$, and on this surface, display the level curves. Under the surface, show the level curves projected onto the plane $z\=15$.

Example 3.1.4

Graph the surface $z\left(x\,y\right)$ defined implicitly by the equation $xy\+z\mathrm{cosh}\left(z1\right)\=1$.

Example 3.1.5

Obtain a contour map for the function $z\left(x\,y\right)$ defined implicitly by the equation $xy\+z\mathrm{cosh}\left(z1\right)\=1$.

Example 3.1.6

Obtain plane sections $x\=c$ for the surface defined by $z\=f\left(x\,y\right)\=103{x}^{2}7{y}^{2}$.

Example 3.1.7

Obtain plane sections $y\=c$ for the surface defined by $z\=f\left(x\,y\right)\=103{x}^{2}7{y}^{2}$.

Example 3.1.8

Obtain plane sections $x\=c$ for the surface defined implicitly by the equation $xy\+z\mathrm{cosh}\left(z1\right)\=1$.

Example 3.1.9

On the domain $0\le x\,y\,z\le 3$, graph the level surfaces $w\=5$ and $w\=15$ for the function $w\=\mathrm{ln}\left(y\+\mathrm{sinh}\left(xz\right)\right)\+y\mathrm{cosh}\left(x\+z\right)$.

Example 3.1.10

On the domain $2\le s\,t\le 2$, graph the surface defined parametrically by the equations$x\=3tplus;{s}^{2}comma;yequals;5s{t}^{2}comma;zequals;{s}^{2}plus;{t}^{2}$.

Example 3.1.11

On the domain $2\le s\,t\le 2$, obtain a contour map for the surface defined parametrically by the equations $x\=3tplus;{s}^{2}comma;yequals;5s{t}^{2}comma;zequals;{s}^{2}plus;{t}^{2}$.${}$



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