Problem 17.6 on
Contour Plots

by
K. D.  Stroyan
University of Iowa

copyright 1997 by Academic Press, Inc. - All rights reserved.

Special Mathematica Functions used in this NoteBook

Notebook Overview

Run this Notebook after you have drawn the explicit surfaces from text Problem 17.6 using the prepared NoteBook ExplicitSurfaces.ma  Leave your solution to the ExplicitSurface.ma homework active on the screen or re-open it.  We want you to compare the surfaces with their contour plots and have prepared the contour plots to save typing time.  However, to keep things interesting, we have hidden wrong plots in the NoteBook.

Move the windows for ContourPlot.ma and your solved homework side by side on your workstation so you can compare the Explicit and Contour graphs.  (If your workstation has a small screen, you may have to flip-flop back and forth between the two open NoteBooks.)

References to the Text

This NoteBook goes with Problem 17.6 and Exercises 18.2.1, 2, 3, 4.

Problem 17.6
Plots For Comparison to Your Explicit Surfaces

Use this Mathematica Notebook along with ExplicitSurfaces.ma to plot the functions below.  Compare the surface plots with their topographical maps.  We typed the formulas with Mathematica's InputForm, rather than StandardForm.  (We have hidden some wrong contour plots  in the Notebook to keep you on your toes.)

Function a

z := x y^2 + x^3 y

ContourPlot[z, {x, -3.2, 3.2}, {y, -3.2, 3.2}];

Function b

z := Sin[x]^3 + Cos[y]^4

ContourPlot[z, {x, -3.2, 3.2}, {y, -3.2, 3.2}];

Function c

z := Sin[x ] Cos[x y]

ContourPlot[z, {x, -2, 2}, {y, -2, 2}];

Function with Sqrt as in ExplicitSurfaces.ma

This is a special case for you to observe.

z := Sqrt[1 - x^2 - y^2]

ContourPlot[z, {x, -2, 2}, {y, -2, 2}];

Function d

z := Sqrt[ 1 - x^2 - y^2 ] Cos[ 2 Pi x y ]

ContourPlot[z, {x, -2, 2}, {y, -2, 2}];

Function e

z := Sqrt[x^2 + Sin[y]]

ContourPlot[z, {x, -2, 2}, {y, -2, 2}];

Function f

z := 1/Sqrt[x^2 + y^2]

ContourPlot[z, {x, -2, 2}, {y, -2, 2}];

Function g

z := (2 x^2 + 3 y^2) Exp[-(x^2+y^2)]

ContourPlot[z, {x, -2, 2}, {y, -2, 2}];

Function h

z := 2 x y Exp[-(x^2/2 + y^2/8)]

ContourPlot[z, {x, -4, 4}, {y, -4, 4}];

Function i

z := ArcTan[x^2 y^3]

ContourPlot[z, {x, -2, 2}, {y, -2, 2}];

Function j

z := Log[x^2 + y^4]

ContourPlot[z, {x, -2, 2}, {y, -2, 2}];

Special Note

You don't have to save this NoteBook.  Just put a message in the Explicit Surface homework about the incorrect contour plot.

Exercise 18.2.3

For each of the functions in Exercise 18.2.1, find one X = (x,y) point where the graph has a horizontal tangent plane, if there is one.  How do you locate these points on a contour plot?  You can compare your work in the PartialD and ExplicitSurfaces NoteBooks with these plots to help answer this question.