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CalcPlot3D, an Exploration Environment for Multivariable Calculus - Level Surfaces

Author(s): 
Paul Seeburger (Monroe Community College)

It is difficult to draw many interesting level surfaces by hand, so I generally have my students use CalcPlot3D to do most of the work for this type of exercise. There are actually two ways to enter and graph the level surface equations for a particular function of three variables in CalcPlot3D:

  1. Solve each equation for z in terms of x, y, and C and enter the level surface using one or two functions of x and y, or
  2. Graph the level surface equation by Adding an Implicit Surface from the Graph menu and entering the equation for the level surface in the dialog box there.

If the surfaces are complicated enough, you may not have a choice. If you are not able to solve for z, you will need to use the Implicit Surface option.

Here is an example I use in class shown both ways. \[ f(x,y,z) = z^2 - x^2 + y^2 \]

Setting \(f(x,y,z)=z^2-x^2+y^2=C\), we obtain the following equations if we solve for z.

Level surface with C=2
Level surface with C=-2
C = 2 C = -2

\(z=\sqrt{C+x^2-y^2}\)
\(z=-\sqrt{C+x^2-y^2}\)

For \(C=2\) we enter:

z=sqrt(2+x^2-y^2) in Function 1, and
z=-sqrt(2+x^2-y^2) in Function 2.

For \(C=-2\) we enter:

z=sqrt(-2+x^2-y^2) in Function 1, and
z=-sqrt(-2+x^2-y^2) in Function 2.

 

We can obtain the following graphs of these surfaces by graphing the implicit equations.

Level curve of z^2-x^2+y^2 = 2 Level curve of z^2-x^2+y^2=-2
\(z^2-x^2+y^2=2\) \(z^2-x^2+y^2=-2\)

\( z^2-x^2+y^2=2\) and \(z^2-x^2+y^2 = -2 \)

These equations will be entered as:

 z^2 - x^2 + y^2 = 2
 z^2 - x^2 + y^2 = -2

 

 

 

 

 

Click here to open the CalcPlot3D applet in a new window.

Click here to open a pdf file which contains the instructions for the activity.

Paul Seeburger (Monroe Community College), "CalcPlot3D, an Exploration Environment for Multivariable Calculus - Level Surfaces," Loci (November 2011), DOI:10.4169/loci003781

Dummy View - NOT TO BE DELETED