Answer to Question #131907 in Differential Equations for VIRENDRA BALKI

Question

Define Index of a stationary point with example. 

Answer 1

Index of a stationary point of the vector field is a topological characteristic of isolate stationar point. It defines as a degree of Gaussian mapping in this point.

Isolate stationar point is the point in some punctured neighbourhood, where function is monosemantic and analityc, and in this point either not given or not differentiable.

Gaussian mapping is same as spherical mapping.

Example:

Forthe non-degenerate point:

$\Delta=det(\frac{d\xi^{\alpha}}{dx^{\beta}})\neq0$, when $x=x_0$

This point is always isolate and it's index equals to the sign of the determinant $\Delta$.