Introduction:
The hyperbolic paraboloid
is often called a saddle surface, because a person could sit on it comfortably.
A monkey, however, would run into trouble because he would have nowhere
to put his tail! The above surface is called a monkey saddle, because
it has a convenient dip in the back to accommodate the monkey's tail.
The monkey saddle is in fact a special case of the generalized
monkey saddle.
Definition:

Properties:
Partial Derivatives:
Because z is a function of x and y, we can take partial derivatives:


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Tangent Planes:
At u = u0, v = v0,
the tangent plane to the surface is parameterized by:

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Infinitesimal Area:
The infinitesimal area of a patch on the surface is given by

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Gaussian Curvature:

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Gaussian curvature of the surface.
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Surface colored by Gaussian curvature.
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Note: The jagged edge near the minimum in the graph
of Gaussian curvature is not a feature of the graph; rather, it is
an artifact of the plotting process. |
Mean Curvature:

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Mean curvature of the surface.
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Surface colored by Mean curvature.
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