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Chapter 3: Functions of Several Variables
Section 3.3: Quadric Surfaces
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Example 3.3.7
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Put the equation into standard form for a quadric surface, identify the surface, draw its graph, and discuss the nature of the level curves and plane sections.
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Solution
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Mathematical Solution
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Figures 3.3.7(a, b) contain a graph of the surface defined by the given equation,
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whose standard form is
obtained by completing the square in and . The standard form is the equation of an elliptic cylinder.
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The point would be the center of the ellipse in the -plane.
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The level curves, drawn on the surface of the cylinder, are ellipses with semi-major axis 3 and semi-minor axis 2.
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The cross sections are vertical lines in the -plane.
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The cross sections are pairs of vertical lines in the -plane.
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=
=
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Figure 3.3.7(a) Elliptic cylinder with cross sections
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=
=
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Figure 3.3.7(b) Elliptic cylinder with cross sections
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Maple Solution - Interactive
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Obtain the standard form
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Control-drag the given equation.
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Context Panel: Manipulate Equation
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Check the "Show steps stacked vertically" box.
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Click the "Complete the square" button.
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Add to both sides and multiply both sides as per the actions shown in the figure below.
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Click the "Return Steps" button.
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Graph via the Context Panel
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Control-drag the equation.
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Context Panel: Plots≻3-D ImplicitPlot≻x, y, ?
Style≻Surface Contour
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Alternative, use the
to obtain the equivalent of the surfaces in Figures 3.3.7(a, b)
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Click the "Interactive Plot Builder" button.
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On the main panel, click "Edit Functions", thereby launching the "Specify Expressions" dialog.
In the Variables section, click Add, and enter as an additional variable. See Figure 3.3.7(d).
After clicking the Add button, the bottom pane will be empty. Click OK.
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Set the ranges as per Figure 3.3.7(c).
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Press the Options button. Set the options as per Figure 3.3.7(f).
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Press the Advanced Settings button in the Axes section; adjust the tickmarks as per Figure 3.3.7(e).
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Figure 3.3.7(c) Main panel of Interactive Plot Builder
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Figure 3.3.7(d) Settings for tickmarks
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Figure 3.3.7(e) Settings for tickmarks
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Figure 3.3.7(f) Options panel, Interactive Plot Builder
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Maple Solution - Coded
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Define so that the graph of is a quadric surface
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Complete the square and put into standard form
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Obtain the equivalent the surface in Figures 3.3.7(a, b)
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<< Previous Example Section 3.3
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