Chapter 3: Functions of Several Variables
Section 3.1: Functions and Their Graphs
Example 3.1.7
Obtain plane sections y=c for the surface defined by z=fx,y=10−3 x2−7 y2.
Solution
Maple Solution - Interactive
The simplest way to obtain graphs of plane sections is to invoke the tutor, which can provide one or more cross sections, or an animation of a single cutting plane moving through a surface forming cross sections as it goes.
Figure 3.1.7(a) is an image of the tutor adjusted to show the single cross section at y=2. Note that the default axis style boxed has been changed to frame. Figure 3.1.7(b) animates the cutting plane y=c as it generates cross sections through the surface z=fx,y.
Figure 3.1.7(a) Cross Section tutor
Figure 3.1.7(b) Animation: cross sections x=c
Maple Solution - Coded
The CrossSection command in the Student MultivariateCalculus package is at the heart of the Cross Section tutor. Table 3.1.7(a) lists two instantiations of this command, the first generating a single cross section at y=2; the second, an animation of one cutting plane generating some 25 cross sections in x∈−3,3.
CrossSection10−3 x2−7 y2,y=2,x=−4..4,y=−4..4,output=plot
CrossSection10−3 x2−7 y2,y=−3..3,x=−4..4,y=−4..4,output=animation,planes=25
Table 3.1.7(a) Instantiations of the CrossSection command
The purist who deigns to construct the image of a cross section from Maple primitives must draw the surface using the plot3d command, the cutting plane using either plot3d parametrically or implicitplot3d, and the curve of intersection using the spacecurve command. Then, all three images must be joined with the display command. Table 3.1.7(b) contains this construction for the plane section at y=1.
p1≔plot3d10−3 x2−7 y2,x=−4..4,y=−4..4,style=surface,transparency=.3:p2≔plot3dx,1,z,x=−4..4,z=−140..10,style=surface,transparency=.3:p3≔plots:-spacecurvex,1,3−3 x2, x=−4..4,color=black,thickness=3:plots:-displayp1,p2,p3,labels=x,y,z
Table 3.1.7(b) Constructing a cross section from primitives
Clearly, Table 3.1.7(b) has been included to emphasize the utility of the CrossSection command and its companion tutor.
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