Functions of two variables

Here is a simple function of two variables

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Evaluated at a few points:

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The following plots our graph for values of x and y lying in the square: -1<x<1 and -1<y<1:

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The BoxRatios→Automatic option makes each axis appear to be at the same scale

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We can use the option to change our point of view:

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Here

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The Coutours option allows us to plot more level sets

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Here we color the contours with colors

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Here's an example where we render some pretty pictures with Mathematica

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The larger we make PlotPoints, the smoother the surface will appear, but the longer it will take to draw
Mesh->False takes away the black grid lines from our surface

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PlotPoints->100 makes the level sets appear smoother

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We can specify which contour lines to plot with the Contours Option
In the folowing plot, we choose level sets correspoinding to F[x,y]=.5, .75, and 1

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Contours can also be used to specify the number of contour lines to plot as below.
We also have increased PlotPoints which makes the level curves more smooth.
ContourLines->False removes the black level set lines from the image.

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Created by Mathematica  (September 20, 2004)