View source for Saddle point

A point on a smooth surface such that the surface near the point lies on different sides of the tangent plane. If a point on a twice continuously-differentiable surface is a saddle point, then the [[Gaussian curvature]] of the surface at the point is non-positive. A saddle point is a generalization of a [[hyperbolic point]].



====Comments====
A surface all of whose points are saddle points is a [[saddle surface]].

A saddle point of a differentiable function $f : M \to \mathbf{R}$ is a point $x$ of the differentiable manifold $M$ which is critical, i.e. $\mathrm{d} f (x) = 0$, non-degenerate, i.e. the Hessian matrix $\left({ \partial^2 f / \partial x^i \partial x^j }\right)$ is non-singular, and such that $x$ is not a local maximum or a local minimum, i.e. the Hessian matrix is indefinite. Thus, a non-degenerate critical point of $f : M \to \mathbf{R}$ is a saddle point if its index (the number of negative eigenvalues of the Hessian matrix at that point) is $\ne 0,\,\dim M$. (The index does not depend on the local coordinates chosen.) The graph of a real-valued function of two variables $f : \mathbf{R}^2 \to \mathbf{R}$ near a saddle point looks like a saddle. See also [[Saddle point in game theory]].

A [[saddle]] of a differential equation on $\mathbf{R}^2$ is also often called a saddle point of that differential equation. More generally, given a dynamical system $\dot x = f(x)$ on $\mathbf{R}^n$ (or on a differentiable manifold) one considers the eigenvalues of $D F(x_0)$ at an equilibrium point $x_0$. If both positive and negative real parts occur, $x_0$ is called a saddle, a saddle point or, sometimes, a Poincaré saddle point.

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  M.W. Hirsch,   S. Smale,   "Differential equations, dynamical systems, and linear algebra" , Acad. Press  (1974)  pp. 190ff  {{MR|0486784}} {{ZBL|0309.34001}} </TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  D.R.J. Chillingworth,   "Differential topology with a view to applications" , Pitman  (1976)  pp. 150ff  {{MR|0646088}} {{ZBL|0336.58001}} </TD></TR>
</table>

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