Chapter 7.2, Problem 5E

Interpretation Introduction

Interpretation:

Let $\dot{\text{x}}\text{ = f}\left(\text{x, y}\right)\text{,}\dot{\text{y}}\text{ = g}\left(\text{x, y}\right)$ be a smooth vector field defined on the phase plane. Show that if this is a gradient system, then $\frac{\text{df}}{\text{dy}}\text{ = }\frac{\text{dg}}{\text{dx}}$

Concept Introduction:

The system can be written in the form $\dot{\text{x}}\text{= -}\nabla \text{V}$ for some continuously differential, single-valued scalar function $\text{V(x)}$. Such a system is called a gradient system with potential function. $\text{V(x)}$

Closed orbit is impossible in the gradient system.