Chapter 3: Functions of Several Variables
Section 3.1: Functions and Their Graphs
Example 3.1.11
On the domain −2≤s,t≤2, obtain a contour map for the surface defined parametrically by the equations
x=3 t+s2,y=5 s−t2,z=s2+t2
Solution
Maple Solution - Interactive
The surface is defined parametrically via equations. Maple's tools for graphing a surface defined parametrically require a list of three expressions. The required graph can be obtained interactively in a syntax-free way by first constructing the list of three expressions. Then the Plot Builder can be invoked to draw a "3-D plot (parametric)".
Control-drag the sequence of equations. Context Panel: Substitute Into≻x,y,z
Context Panel: Plots≻Plot Builder≻3-D plot (parametric) Edit In "Basic Options", set −2≤s,t≤2 and set style to surfacecontour In "Axes and Text", set the label for axis[1] to x, and for axis[2] to y In "3-D Options", set the orientation to −90,0,0 and set "lighting model" to none
x=3 t+s2,y=5 s−t2,z=s2+t2→substitute intos2+3t,−t2+5s,s2+t2→
The resulting graph is Figure 3-1-11(a).
Maple Solution - Coded
The graph in Figure 3.1.11(b) is drawn with Maple's plot3d command.
The first argument to plot3d must be a list of expressions when the surface to be drawn is given parametrically.
The commands for obtaining Figure 3.1.11(b) via the plot3d command appear in Table 3.1.11(a). The eval command is first used to "convert" the sequence of equations into a list of expressions.
Specifying the number of tickmarks per axis leads to less clutter; extinguishing external light impinging on the surface adds to the illusion that the contours have been projected to the xy-plane.
Figure 3.1.11(b) Graph of parametric surface
q≔evalx,y,z,x=3 t+s2,y=5 s−t2,z=s2+t2:plot3dq,s=−2..2,t=−2..2,labels=x,y,,tickmarks=5,5,10,orientation=−90,0,0,lightmodel=none,style=surfacecontour
Table 3.1.11(a) Syntax for obtaining Figure 3.1.11(b)
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