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Encyclopedia > Saddle point
Plot of y = x3 with a saddle-point at (0,0).
Plot of y = x3 with a saddle-point at (0,0).

In mathematics, a saddle point is a point of a function (of one or more variables) which is both a stationary point and a point of inflection. Since it is a point of inflection, it is not a local extremum. Image File history File links I created this graph using software I wrote and grant full license to anyone to use, copy, and distribute. ... Image File history File links I created this graph using software I wrote and grant full license to anyone to use, copy, and distribute. ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... The word point can refer to: a location in physical space a unit of angular measurement; see navigation point is a typographic unit of measure in typography equal inch or sometimes approximated as inch; on computer displays it should be equal to point in typography if the correct display resolution... In general, a function is part of an answer to a question about why some object or process occurred in a system that evolved or was designed with some goal. ... In computer science and mathematics, a variable is a symbol denoting a quantity or symbolic representation. ... In mathematics, particularly in calculus, a stationary point is a point on the graph of a function where the tangent to the graph is parallel to the x-axis or, equivalently, where the derivative of the function equals zero (known as a critical number). ... In mathematics, particularly in calculus, a stationary point is a point on the graph of a function where the tangent to the graph is parallel to the x-axis or, equivalently, where the derivative of the function equals zero (known as a critical number). ... A graph illustrating local min/max and global min/max points In mathematics, a point x* is a local maximum of a function f if there exists some ε > 0 such that f(x*) ≥ f(x) for all x with |x-x*| < ε. ...


For a function of a single variable, such a point is one where the first derivative is zero, and the second derivative changes sign. For example, the function y = x3 has such a point at the origin. The derivative in mathematics (specifically, differential calculus) is a quantity that measures, on continuous functions, the limit of a rate of change, , as approaches 0. ... In mathematics, the origin of a coordinate system is the point where the axes of the system intersect. ...

Saddle point in the graph of z=x²-y²
Saddle point in the graph of z=x²-y²

For a function of two or more variables, the surface at a saddle-point resembles a saddle that curves up in one ore more directions, and curves down in one or more other directions (like a mountain pass). In terms of contour lines, a saddle point can be recognised, in general, by a contour that appears to intersect itself. For example, two hills separated by a high pass will show up a saddle point, at the top of the pass, like a figure-eight contour line. Wikipedia does not have an article with this exact name. ... Wikipedia does not have an article with this exact name. ... In mathematics, a surface is a two-dimensional manifold. ... Tack is any of the various accessories worn by horses in the course of their use as domesticated animals. ... In a range of hills, or especially of mountains, a pass (also gap, notch, col, saddle, bwlch or bealach) is a lower point that allows easier access through the range. ... Contour map A contour line (also level set, isogram or isarithm) for a function of two variables is a curve connecting points where the function has a particular value. ... In mathematics, a lemniscate is a type of curve described by a Cartesian equation of the form: Graphing this equation produces a curve similar to . ...


More formally, given a real function F(x,y) of two real variables, the Hessian matrix H of F is a 2×2 matrix. If it is indefinite (neither H nor −H is positive definite) then in general it can be reduced to the Hessian of the function In mathematics, the Hessian matrix is the square matrix of second partial derivatives of a scalar-valued function. ... In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ...

x2y2,

at the point (0,0). This function has a saddle point there, curving up along the line y = 0 and down along the line x = 0.


In fact if H is a non-singular matrix (general case) and F is smooth enough, this is the correct local model for a stationary point of F that is not a local maximum nor a local minimum. If H has rank < 2 one cannot be certain in the same way about the local behaviour. In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ... A graph illustrating local min/max and global min/max points In mathematics, a point x* is a local maximum of a function f if there exists some ε > 0 such that f(x*) ≥ f(x) for all x with |x-x*| < ε. ... A graph illustrating local min/max and global min/max points In mathematics, a point x* is a local maximum of a function f if there exists some ε > 0 such that f(x*) ≥ f(x) for all x with |x-x*| < ε. ...


In physics, wet electrons are a saddle point between electrons in liquid and electrons in solid. This article needs to be cleaned up to conform to a higher standard of article quality. ...


See also


  Results from FactBites:
 
Saddle point - Wikipedia, the free encyclopedia (409 words)
In the most general terms, a saddle point for a smooth function (curve, surface or hypersurface) is a point such that the curve/surface/etc. in the neighborhood of this point lies on different sides of the tangent at this point.
A saddle point is a point of a function (of one or more variables) which is both a stationary point and a point of inflection.
A saddle point is an element of the matrix which is both the smallest element in its row and the largest element in its column is called saddle point.
Saddle at AllExperts (923 words)
To non-horsemen, the major distinguishing feature of an English saddle is its lack of a horn and its panels: a pair of pads attached to the underside of the seat and filled with wool, foam, or air.
However, it was not until 700-800 BCE that the predecessor to the modern saddle evolved, thought to have been developed by the Assyrians, and consisting of a simple cloth fastened onto the horse with a girth.
Saddles were greatly improved upon during the Middle Ages, as knights needed something more supportive and stronger, and the resulting saddle had a higher cantle (to prevent the rider from being unseated backwards by a lance) and pommel (to protect the rider's stomach from the opposer's lance) and was built on a wooden frame.
  More results at FactBites »

 
 

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