Periodic sequence: Difference between revisions

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[[Image:Monkey Saddle Surface (Shaded).png|thumb|The monkey saddle|300px]]
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In [[mathematics]], the '''monkey saddle''' is the [[surface]] defined by the equation


:<math> z = x^3 - 3xy^2. \, </math>
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It belongs to the class of [[saddle surface]]s and its name derives from the observation that a [[saddle]] for a [[monkey]] requires three depressions: two for the legs, and one for the tail. The point (0,0,0) on the monkey saddle corresponds to a [[critical point (mathematics)|degenerate critical point]] of the function ''z''(''x'',''y'') at (0, 0). The monkey saddle has an isolated [[umbilical point]] with zero [[Gaussian curvature]] at the origin, while the curvature is strictly negative at all other points.
 
To show that the monkey saddle has three depressions, one can write the equation for ''z'' using [[complex number]]s as
:<math> z(x,y) = \operatorname{Re} (x+iy)^3.</math>
It follows that ''z''(''tx'',''ty'') = ''t''<sup>3</sup> ''z''(''x'',''y'') for ''t'' ≥ 0, so the surface is determined by ''z'' on the [[unit circle]]. Parametrizing this by ''e''<sup>''i''φ</sup>, with φ ∈ [0, 2π), one can see that on the unit circle, ''z''(φ) = cos 3φ, so ''z'' has three depressions. By replacing 3 with any integer ''k'' ≥ 1, one can create a saddle with ''k'' depressions.
 
== Horse saddle ==
 
The term ''horse saddle'' is used, in contrast to monkey saddle, to designate a [[saddle point]] that is a [[minimax]], that is to say a local minimum or maximum depending on the intersecting plane used. The monkey saddle has just a [[point of inflection]]. To see this, consider a line ''y'' = ''kx''. Along this direction the surface becomes simply ''z'' = (1 &minus; 3''k''<sup>2</sup>)''x''<sup>3</sup>, lacking any critical points.
 
== See also==
* [[Saddle point]]
*[[Saddle surface]]
 
==External links==
* {{MathWorld | urlname=MonkeySaddle | title=Monkey Saddle}}
 
[[Category:Multivariable calculus]]
[[Category:Surfaces]]

Latest revision as of 00:06, 29 November 2014

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