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EXAMPLE 9 In cylindrical coordinates, the equation z = r represents the cone z =
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x 2 + y2 .
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RevolutionPlot3D[r,{r,0,1},BoxRatios 1]
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Point P has spherical coordinates (r, q, f) where r is the distance from P to the origin, q is the angle formed by the positive x-axis and the line connecting the origin with the projection of P in the x-y plane, and f is the angle formed by the positive z-axis and the line connecting P with the origin. The Mathematica command SphericalPlot3D allows the construction of surfaces given in spherical coordinates. Special Note: When dealing with spherical coordinates, Mathematica s convention is to interchange the roles of q and f from that which is used in many standard calculus textbooks. The description of the command SphericalPlot3D described in the following, although different from the description in Mathematica s documentation files, agrees with these conventions.
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SphericalPlot3D[q, e, p] generates a complete plot of the surface whose spherical radius, , is defined as a function of and . SphericalPlot3D[[q, {e, emin, emax}, {p, pmin, pmax}] generates a plot of the surface whose spherical radius, , is defined as a function of and over the intervals min max and min max.
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EXAMPLE 10 In spherical coordinates, r = 1 represents the unit sphere.
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SphericalPlot3D[1, {e, 0, o}, {p, 0, 2o}]
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Three-Dimensional Graphics
SOLVED PROBLEMS
5.1 Plot the graph of the function e x
SOLUTION
2 y2
above the rectangle 2 x 2, 2 y 2.
Plot3D[Exp[ x2 y2], {x, 2, 2}, {y, 2, 2}]
1.0 2 0.5 0.0 2 1 0 1 2 2 1 0 1
5.2 Show the intersection of the two paraboloids f ( x , y) = x 2 + y 2 and g( x , y) = 16 x 2 y 2 above the square 3 x 3, 3 y 3.
SOLUTION
Plot3D[{x2 + y2, 16 x2 y2}, {x, 3, 3}, {y, 3, 3}, BoxRatios 1]
2 0 2
5 0 2 0 2
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5.3 Obtain a graph of the saddle-shaped hyperboloid z = x 2 y 2 , 5 x 5, 5 y 5 in a cubic box. Draw the graph with and without a surface mesh.
SOLUTION (Graphs are placed side by side for comparison purposes.)
Plot3D[x2 y2, {x, 5, 5}, {y, 5, 5}, BoxRatios 1] Plot3D[x2 y2, {x, 5, 5}, {y, 5, 5}, BoxRatios 1, Mesh False]
5 0 5 20 5 20 0
20 5 0 5
20 5 0 5
5.4 Draw the graph of the function f ( x , y) = sin x sin y for 2 x, y 2 . Label the x and y axes in terms of .
SOLUTION
Plot3D[Abs[Sin[x]Sin[y]], {x, 2 o, 2 o}, {y, 2 o, 2 o}, Ticks {{ 2 o, o, 0, o, 2 o},{ 2 o, o, 0, o, 2 o}, {0, 1}}]
1 2 0 2 0 2 2 0
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5.5 Draw the graph of the surface z = 1 x 2 y 2 for 1 x, y 1. Do not draw axes or a surrounding box.
SOLUTION
Plot3D[Abs[1 x2 y2], {x, 1, 1}, {y, 1, 1}, Axes False, Boxed False]
5.6 Graph the intersection of the paraboloid z = x 2 + y 2 with the plane y + z = 12. Obtain a front view and a side view.
SOLUTION (Graphs are placed side by side for easier comparison.)
paraboloid = Plot3D[x2 + y2, {x, 5, 5}, {y, 5, 5}]; plane = Plot3D[12 y, {x, 5, 5}, {y, 5, 5}]; Show[paraboloid, plane, BoxRatios 1, PlotRange {0, 20}, PlotLabel "Default View"] Show[paraboloid, plane, BoxRatios 1, PlotRange {0, 20}, ViewPoint Front, PlotLabel "Front View"] Show[paraboloid, plane, BoxRatios 1, PlotRange {0, 20}, ViewPoint Left, PlotLabel "Left View"]
Default View 5 0 5 20 15 10 5 0 5 0 5 5 5 0 5 0 5 0 5 5 5 0 0 5 5 5 0 20 15 15 10 10 Front View Left View 20
x = cos at 5.7 Sketch the space curves defined by y = sin bt z = sin ct (i) a = 5, b = 3, c = 1; (ii) a = 3, b = 3, c = 1; These curves are known as Lissajous curves.
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