Chapter 6.8, Problem 13E

Interpretation Introduction

Interpretation:

If $\dot{\text{x}}\text{ = f}\left(\text{x, y}\right)$, $\dot{\text{y}}\text{ = g}\left(\text{x, y}\right)$ is a smooth vector field on the plane, C is a simple closed curve that does not pass through any fixed points, $\varphi {\text{ = tan}}^{\text{-1}}\left(\frac{\dot{\text{y}}}{\dot{\text{x}}}\right)$ then $\text{d}\varphi \text{ = }\left(\frac{\text{f dg - g df}}{{\text{f}}^{\text{2}}{\text{ + g}}^{\text{2}}}\right)$ is to be showed and the integral formula ${\text{I}}_{\text{c}}\text{ = }\frac{\text{1}}{\text{2π}}{\displaystyle {\oint }_{\text{C}}\left(\frac{\text{f dg - g df}}{{\text{f}}^{\text{2}}{\text{ + g}}^{\text{2}}}\right)}$ is to be derived.

Concept Introduction:

Index theory provides the global information about the phase portrait and fixed points.

The index of the closed curve $C$ is an integer which measures the winding of the vector field on closed curve C.

The index of the closed curve with respect to vector field can be mathematically expressed as

${\text{I}}_{\text{c}}\text{ = }\frac{\text{1}}{\text{2π}}{\left[\varphi \right]}_{\text{C}}$

where ${\text{I}}_{\text{c}}$ is the net number of counterclockwise revolutions that are made by the vector field, ${\left[\varphi \right]}_{\text{C}}$ is the net change in $\varphi$ around $\text{C}$.