Problem 17.6 on
Explicit Surfaces

by
K. D.  Stroyan
University of Iowa

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

Notebook Overview

This Mathematica Notebook is to help you  make graphs of the functions given in text Problem 17.6 and followed up in Exercises 18.2.1, 2, 3, 4.  We have started the first few to help you learn how to use the Plot3D[] function.

Functions are defined as shown below.  The underline after the variables indicates that it is a rule about x and y.  Multiplication is indicated by a space.  The variable  xy  is a single variable, but  x y  means  x  times  y  (note the space between x and y).  Powers are denoted by the  ^  symbol as shown in the first example below.

Move the mouse so that the insert symbol is in the next cell, click, and then press the <Enter> key at the far right of the keyboard (near the number pad.)  This will command Mathematica to draw the explicit graph  z = f(x,y)  for the first function of the exercise.  The range of the plot is:  -3.2 < x < 3.2  and  -3.2 < y < 3.2.  Draw the graph now.

References to the Text

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

Problem 17.6

Use this NoteBook to plot all of the functions in this exercise. Compare the surface plots with their topographical maps from the Mathematica Notebook ContourPlots.

Function a

<Enter> this computation:

f[x_, y_] := x (y^2) + (x^3) yPlot3D[f[x, y], {x, -3.2, 3.2}, {y, -3.2, 3.2}];

Function b

The second function is defined below for you.  Notice that Mathematica knows the sine and cosine functions, but that they must be typed with the notation  Sin[x], Cos[y],  that is, the first letter is capitalized and the argument goes between square brackets.  The list of symbols  "sin(x)"  is treated simply as a new variable name.  Mathematica does not recognize it as having anything to do with the sine function.  It is also prejudiced against round brackets or parentheses ( and )  in functions, though you must use them in algebraic expressions such as

(x + y)/(x - y)

Move the mouse to the next cell, click, and command Mathematica to plot the surface.

f[x_, y_] := (sin^3) x + (cos^4) y    Plot3D[f[x, y], {x, -3.2, 3.2}, {y, -3.2, 3.2}]

Function c

Highlight the whole command cell above containing the function

f[x_, y_] := Sin[x]^3 + Cos[y]^4

(Not the cell with the surface you just drew, the one with the instructions to draw it.)  You first put the mouse at the beginning of the cell before  f[x_,y_], then press down the mouse button and hold it while you move to the end of the cell by  ,3.2}].  This is called "dragging" the mouse.  This will highlight the whole cell.

Once the cell is highlighted, copy it by holding down the <Command> key and pressing the <c> key.  You won't see anything happen, but move the mouse so the insert bar is in the blank window just below here, click, and then press <Command> + <v> which is the "paste" command.  This should give you a copy of the previous comand window.  Edit it so you have the function

f(x,y)  = Sin[x*y]*Cos[x*y]

(Fix the syntax so that Mathematica can understand you!)
and plot command

Plot3D[f[x,y], {x,-2,2}, {y,-2,2}]

Then press <Enter> and draw the graph of the third function of the homework.

f[x_,y_] := Sin[x*y]*Cos[x*y]
Plot3D[???

A Function with a Square Root

The square root function is used in the next cell.  You type it with "Sqrt[ ]" or the radical symbol from the Basic Input Palette.  Graph the function in the next cell now and explain the error messages. (Type your explanation in the blank text cell below the graph.)

f[x_, y_] := (Sqrt[1 - (x^2) - (y^2)])Plot3D[f[x, y], {x, -1, 1}, {y, -1, 1}]

Type your explanation here:

Function d

Now copy the cell above into the next command cell and edit it to plot the function

f(x,y)  = SquareRoot(1 - x^2 - y^2) . cos 2πxy

Don't forget that space (or * ) means times... and Mathematica uses [.] for functions and  Sin,...

Remaining Functions

Finish the assignment with the functions:

e)  z = (Sqrt[(x^2) + Sin[(y^2)]])

f) z = (1/(Sqrt[(x^2) + (y^2)]))

g)  z = (2 (x^2) + 3 (y^2)) Exp[-((x^2) + (y^2))]
	  or  z = (2 (x^2) + 3 (y^2)) (e^(-((x^2) + (y^2))))

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

i)  z = ArcTan[(x^2) (y^3)]

j)  z = Log[(x^2) + (y^4)]

Simply copy, paste and edit as needed.  Be careful to plot your functions over interesting domains of the independent variables.

Function ?

Special Message

Save this NoteBook on your account, but leave it open and also open the ContourPlots NoteBook.  The second part of this assignment is to compare your explicit surfaces with their contour plots.

Exercise 18.2.3

For each of the functions in Exercise 18.2.1 - plotted above, find one X or (x,y) point where the graph has a horizontal tangent plane, if there is one.  Use this NoteBook to help see where to look for the solution of the appropriate condition on the partial derivatives you computed in Exercise 18.2.2 using the PartialD NoteBook.