Chapter 3: Functions of Several Variables
Section 3.1: Functions and Their Graphs
Example 3.1.2
Obtain a contour map of the function z=fx,y=10−3 x2−7 y2.
Solution
Maple Solution - Interactive
A simple way to obtain the contours shown in Figures 3.1.2(c-d) is to use the Context Panel to obtain a graph of the surface, modify the style to show level curves, and then to rotate the surface so the viewpoint is directly above.
Control-drag the rule for f.
Context Panel≻Plots≻3-D Plot≻x,y
Bring up the Context Menu directly on the graph (secondary click) Style → Surface Contour Use the mouse to rotate the surface so the viewpoint is from directly above.
10−3 x2−7 y2→
Alternatively, use the Plot Builder.
Control-drag the rule for the function.
Context Panel: Plot Builder≻2-D contour plot 2-D Options: Scaling≻constrained
Figures 3.1.2(a-d) are obtained with the . Figures 3.1.2(a) and 3.1.2(d) are "contour maps" for the surface defined by z=fx,y. It certainly helps to see the contours on the surface itself, as shown in Figure 3.1.2(c). Unless the colors in Figure 3.1.2(b) are enhanced in some way (all set to black?), the image is not as useful as the one in Figure 3.1.2(c).
Figure 3.1.2(a)
Figure 3.1.2(b)
Figure 3.1.2(c)
Figure 3.1.2(d)
Figure 3.1.2(a) is obtained by selecting "2-D contour plot" in the Select Plot section of the main panel of the Plot Builder. (See Figure 3.1.2(e).)
Figure 3.1.2(b) is obtained by selecting "3-D contour plot".
Figure 3.1.2(c) is obtained by selecting "3-D plot" and setting Style to "surface with contour" in the Options panel.
Figure 3.1.2(e)
Figure 3.1.2(d) is obtained by rotating the graph in Figure 3.1.2(c) so its orientation angles are ϑ=−90, ϕ=ψ=0, that is, by rotating the graph so it is viewed from directly above.
Figures 3.1.2(f-h) derive from the tutor. Figures 3.1.2(g-h) show how to adjust the Plot Options and the main panel to obtain ten uniformly spaced planes z=c,c∈−50,10. The intersections of these planes with the surface z=fx,y can be seen all together by pressing the Display button in the tutor. Alternatively, pressing the Animate button yields an animation similar to the one shown in Figure 3.1.2(f).
Figure 3.1.2(f)
Figure 3.1.2(g)
Figure 3.1.2(h)
At this time, it is not yet possible to map specific colors or linestyles onto contours. Table 3.1.2(a) contains a task template that provides the value of f at the point clicked on. However, contour maps generated with the contourplot command now support an automatic hover mechanism that provides function values on the contours in the graph.
Tools≻Tasks≻Browse: Calculus - Multivariate≻Utilities≻Values Along Contours
Values along Contours
fx,y= = c
≤x≤
≤y≤
Grid density:
fx,y=
Table 3.1.2(a) The Values-along-Contours task template
The contourplot command is at the heart of this task template. Hence, there is provision for declaring the number of contours, or a list of function values that determine where the contours are to be drawn. Higher values in the grid-density slider causes the graph to be drawn with more points. Function values on or near a contour will be displayed under the graph when a point in the graph is clicked on with the mouse. Unfortunately, only one value can be displayed at a time, so the information about a function value on a contour is lost as soon as another point is selected.
Maple Solution - Coded
Table 3.1.2(b) shows how to obtain contour maps with the Maple commands contourplot, contourplot3d, CrossSection, and plot3d. The first two commands accept the argument filledregions = true, the effect of which can be seen in Figures 3.1.2(i, j). In all cases, the number of contours can be controlled with the optional parameter contours, which is set equal to a positive integer, or to a list of function values for which a contour is to be drawn.
Figure
Command
plots:-contourplot10−3 x2−7 y2,x=−5..5,y=−5..5
plots:-contourplot3d10−3 x2−7 y2,x=−5..5,y=−5..5
Figures 3.1.2(c, d)
plot3d10−3 x2−7 y2,x=−5..5,y=−5..5,style=surfacecontour
Student:-MultivariateCalculus:-CrossSection10−3 x2−7 y2 , z = 0 .. 32, x = −4 .. 4, y = −4 .. 4, output = plot, planes = 5, axes = frame, caption=
Student:-MultivariateCalculus:-CrossSection10−3 x2−7 y2,z = −50 .. 10, x = −5 .. 5, y = −5 .. 5, z = −50 .. 10, output = animation, planes = 10, axes = frame, caption=,labels=x,y,z
Table 3.1.2(b) Maple commands for obtaining contour maps
The color spectrum used by Maple when the filledregions option is invoked can be controlled by the user with the additional optional parameter coloring, which is set equal to a list of two colors that bound the spectrum.
Figure 3.1.2(i) contourplot with filledregions
Figure 3.1.2(j) contourplot3d with filledregions
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