Chapter 3: Functions of Several Variables
Section 3.1: Functions and Their Graphs
Example 3.1.3
Graph the surface determined by z=fx,y=10−3 x2−7 y2, z∈−10,10, and on this surface, display the level curves. Under the surface, show the level curves projected onto the plane z=−15.
Solution
Figure 3.1.3(a) exhibits the requisite graph.
The Maple code that generates this graph is hidden behind the table cell in which the graph appears. A set of commands that will generate this graph appears in Table 3.1.3(a).
Via the plot3d command, the surface is drawn in such a way that its bottom edge lies in the plane z=−10.
The contour map appearing in green is a projection of the level curves into the plane z=−15, the gap being inserted so that these projections would be visible.
use plots, plottools in module() local p1,p2,p3,f,F,C; f:=10-3*x^2-7*y^2; C:=[-9,-7,-5,-3,-1,1,3,5,7,9]; F:=transform((x,y)->[x,y,-15]); p1:=plot3d(f,x=-sqrt(140/21)..sqrt(140/21),y=-(1/7)*sqrt(-21*x^2+140)..(1/7)*sqrt(-21*x^2+140),style=surfacecontour,contours=C): p2:=contourplot(f,x=-sqrt(140/21)..sqrt(140/21),y=-sqrt(140)/7..sqrt(140)/7,color=green,contours=C): p3:=display([F(p2),p1],view=-15..10,labels=[x,y,z],tickmarks=[5,3,5],axes=frame,orientation=[-60,75,0]); print(p3); end module: end use:
Figure 3.1.3(a) Surface and contour map
A graph such as shown in Figure 3.1.3(a) has to be constructed from several components; there isn't a single Maple command that will accomplish this task.
f≔10−3 x2−7 y2:F≔plottools:-transformx,y→x,y,−15:Y≔solvef=−10,y:C≔−9,−7,−5,−3,−1,1,3,5,7,9:X≔140/21:p1≔plot3df,x=−X..X,y=Y2..Y1,style=surfacecontour,contours=C:p2≔plots:-contourplotf,x=−X..X,y=−140/7..140/7,color=green,contours=C:plots:-displayFp2,p1,view=−15..10,labels=x,y,z,tickmarks=5,3,5,axes=frame,orientation=−60,75,0
Table 3.1.3(a) Maple commands for constructing Figure 3.1.3(a)
Because the contourplot command returns a 2D plot-structure; and plot3d, a 3D plot-structure, without intervention these two graphs could not be merged. Hence, the transform command is used to change each two-dimensional point in the 2D plot-structure to a three-dimensional point, with the third coordinate being z=−15. This action is embedded in the function F, and F is applied to the plot-structure for the contour map of f. To get the bottom of the surface to lie in the plane z=−10, it is graphed with the plot3d command using the domain y∈−Y,Y, where Y=140−21 x2/7, an expression obtained by solving f=−10 for yx. Assigning the list of contour values to the name C is just a device for uncluttering the argument lists for the plotting commands.
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