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[[File:Saddle point.png|thumb|right|200px|A saddle point on the graph of z=x<sup>2</sup>−y<sup>2</sup> (in red)]]
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
[[File:Saddle pt.jpg|thumb|150px|right|Saddle point between two hills (the intersection of the figure-eight <math>z</math>-contour)]]
In [[mathematics]], a '''saddle point''' is a point in the [[domain (mathematics)|domain]] of a [[function (mathematics)|function]] that is a [[stationary point]] but not a [[local extremum]]. The name derives from the fact that the prototypical example in two dimensions is a [[surface]] that ''curves up'' in one direction, and ''curves down'' in a different direction, resembling a [[saddle]] or a [[mountain pass]]. In terms of [[contour line]]s, a saddle point in two dimensions gives rise to a contour that appears to intersect itself.


== Mathematical Discussion ==
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A simple criterion for checking if a given stationary point of a real-valued function ''F''(''x'',''y'') of two real variables is a saddle point is to compute the function's [[Hessian matrix]] at that point: if the Hessian is [[Positive-definite matrix#Indefinite|indefinite]], then that point is a saddle point. For example, the Hessian matrix of the function <math>z=x^2-y^2</math> at the stationary point <math>(0, 0)</math> is the matrix
Registered users will be able to choose between the following three rendering modes:  
: <math>\begin{bmatrix}
2 & 0\\
0 & -2 \\
\end{bmatrix}
</math>
which is indefinite. Therefore, this point is a saddle point. This criterion gives only a sufficient condition. For example, the point <math>(0, 0)</math> is a saddle point for the function <math>z=x^4-y^4,</math> but the Hessian matrix of this function at the origin is the null matrix, which is not indefinite.


In the most general terms, a '''saddle point''' for a [[smooth function]] (whose [[graph of a function|graph]] is a [[curve]], [[surface]] or [[hypersurface]]) is a stationary point such that the curve/surface/etc. in the [[neighborhood (mathematics)|neighborhood]] of that point is not entirely on any side of the [[tangent space]] at that point.
'''MathML'''
[[File:x cubed plot.svg|thumb|150px|The plot of ''y''&nbsp;=&nbsp;''x''<sup>3</sup> with a saddle point at 0]]
:<math forcemathmode="mathml">E=mc^2</math>


In one dimension, a saddle point is a [[Point (geometry)|point]] which is both a [[stationary point]] and a [[Inflection point|point of inflection]]. Since it is a point of inflection, it is not a [[local extremum]].
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


== Other uses ==
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


In [[dynamical systems]], a ''saddle point'' is a [[periodic point]] whose [[stable manifold|stable]] and [[unstable manifold]]s have a [[dimension]] that is not zero. If the dynamic is given by a [[differentiable map]] ''f'' then a point is hyperbolic if and only if the differential of ''&fnof;'' <sup>''n''</sup> (where ''n'' is the period of the point) has no eigenvalue on the (complex) [[unit circle]] when computed at the point.
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


In a two-player [[zero-sum (game theory)|zero sum]] game defined on a continuous space, the [[Nash equilibrium|equilibrium]] point is a saddle point.
==Demos==


A saddle point is an element of the matrix which is both the largest element in its column and the smallest element in its row.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


For a second-order linear autonomous systems, a [[critical point (mathematics)|critical point]] is a saddle point if the [[Characteristic equation (calculus)|characteristic equation]] has one positive and one negative real eigenvalue.<ref>{{harvnb|von Petersdorff|2006}}</ref>


== See also ==
* accessibility:
* [[Saddle-point method]] is an extension of [[Laplace's method]] for approximating integrals
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
* [[Extremum]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
* [[First derivative test]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
* [[Second derivative test]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
* [[Higher-order derivative test]]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
* [[Saddle surface]]
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
* [[Hyperbolic equilibrium point]]
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.
* [[Sion's minimax theorem]]
* [[Mountain pass]]
* [[Max–min inequality]]


== Notes ==
==Test pages ==
<references/>


== References ==
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


* {{citation |last1=Gray |first1=Lawrence F.|last2=Flanigan|first2=Francis J.|last3=Kazdan|first3=Jerry L.|last4=Frank|first4=David H|last5=Fristedt|first5=Bert |title=Calculus two: linear and nonlinear functions |publisher=Springer-Verlag |location=Berlin |year=1990 |pages= page 375|isbn=0-387-97388-5 |oclc= |doi=}}
*[[Inputtypes|Inputtypes (private Wikis only)]]
* {{Citation | last1=Hilbert | first1=David | author1-link=David Hilbert | last2=Cohn-Vossen | first2=Stephan | author2-link=Stephan Cohn-Vossen | title=Geometry and the Imagination | publisher=Chelsea | location=New York | edition=2nd | isbn=978-0-8284-1087-8 | year=1952 }}
*[[Url2Image|Url2Image (private Wikis only)]]
* {{citation|first=Tobias|last=von Petersdorff|url=http://www2.math.umd.edu/~petersd/246/stab.html|chapter=Critical Points of Autonomous Systems|year=2006|title=Differential Equations for Scientists and Engineers (Math 246 lecture notes)}}
==Bug reporting==
* {{citation |last=Widder|first=D. V. |title=Advanced calculus |publisher=Dover Publications |location=New York |year=1989 |pages=page 128 |isbn=0-486-66103-2 |oclc= |doi=}}
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
* {{citation |last=Agarwal|first=A. |title=Study on the Nash Equilibrium (Lecture Notes)|url=http://www.cse.iitd.ernet.in/~rahul/cs905/lecture3/nash1.html#SECTION00041000000000000000}}
 
{{DEFAULTSORT:Saddle Point}}
[[Category:Differential geometry of surfaces]]
[[Category:Multivariable calculus]]
[[Category:Stability theory]]
[[Category:Analytic geometry]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

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