Monkey Saddle

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Monkey Saddle
Field: Calculus
Image Created By: Mathematica
Website: Mathematica

Monkey Saddle

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.


Contents

A More Mathematical Explanation

Note: understanding of this explanation requires: *Calculus

Expressions Defining the Surface

The monkey saddle is defined, in Cartesian coordinates, by the equation:

z(x,y)=x^2 − 3 x y^2

It can also be described by the parametric equations:

x(u,v)=u
y(u,v)=v
z(u,v)=u^2 - 3xy^2


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:

E = 1 + 9(u^2 + v+2)^2
F = -18uv(u^2-v^2)
G = 1+36u^2v^2

And the coefficients of the second fundamental form of the monkey saddle are:

e = \frac{(6u)}{(\sqrt{1+9(u^2+v^2)^2})}
f = \frac{-(6v)}{(\sqrt{1+9(u^2+v^2)^2})}
g = \frac{-(6u)}{(\sqrt{1+9(u^2+v^2)^2})}

Area Element

Thus, the area element of the monkey saddle is given by:

dA=\sqrt{EG-F^2}du \wedge dv = \sqrt{1+9(u^2+v^2)^2}du \wedge dv











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