Monkey Saddle
From Math Images
- This image shows a surface known as a monkey saddle.
Monkey Saddle |
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Contents |
Basic Description
The monkey saddle is a surface in Multivariable Calculus that belongs to the class of saddle surfaces. The surface gets its name from the fact that it has three depressions like a saddle for a monkey, which would require two depressions for the legs and one for the monkey's tail.A More Mathematical Explanation
- Note: understanding of this explanation requires: *Calculus
Expressions Defining the Surface
The monkey saddle is defined, in UNIQ3d7a8b57e3eb7e3-balloon [...]Expressions Defining the Surface
The monkey saddle is defined, in Cartesian coordinates, by the equation:
It can also be described by the parametric equations:
The point (0,0,0) corresponds to a degenerate critical point of the function z(x,y) at (0,0). It is the surface's only stationary point, or point where the derivative of the function is zero. This point is also a saddle point, a point on the surface which is a stationary point, but not an extremum.
Fundamental Forms
The coefficients of the first fundamental form of the monkey saddle are given by:
And the coefficients of the second fundamental form of the monkey saddle are:
Area Element
Thus, the area element of the monkey saddle is given by:
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