13-1-26-stewart.nb

Problem 13-1-26 in Stewart, 5'th ed.

It's pretty darned easy to make a parameterized curve. Here's the one corresponding to our problem.

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$R[t_] = {Sin[t], Cos[t], Sin[t]^2} ;$

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[Graphics:HTMLFiles/index_3.gif]

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Making surfaces is a bit trickier. In this case, the parabolic cylinder is the easier of the two. We let and be free. (Note: I picked nice values for the ranges of these, from to by trial and error. In the curve above, the interval from to was fairly obvious as a choice.)

$RowBox[{parcyl, =, RowBox[{ParametricPlot3D, [, RowBox[{{x, y, x^2}, ,, RowBox[{{, RowBox[{x, ... Box[{-, 1.5}], ,, 1.5}], }}], ,, RowBox[{{, RowBox[{y, ,, RowBox[{-, 1.5}], ,, 1.5}], }}]}], ]}]}]$

[Graphics:HTMLFiles/index_13.gif]

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To plot the cylinder , I'll parameterize the circular part in the usual way and let be free.

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[Graphics:HTMLFiles/index_18.gif]

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Here they are together. The viewpoint is inconvenient, so in the second image we change it from Mathematica's default and look at the figure from above (more or less)..

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[Graphics:HTMLFiles/index_21.gif]

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[Graphics:HTMLFiles/index_24.gif]

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I want to show the curve along with these surfaces but joining a rendering of a one-dimensional curve with that of a surface can lead to "hidden line" problems, so I'll actually use some small spheres along the path of the curve, sort of like a necklace. We first load a package that has a few predefined shapes.

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<<Graphics`Shapes`

A spehere of radius 2, centered at the origin, made with 20 slices longitudinally and 10 latitudinally.

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