Show that if (R) is a gradient system then $\frac{\partial f_i}{\partial x_i}-\frac{\partial f_j}{\partial x_j}=0.$

Question

The dynamical system $\dot{\vec x}=f(\vec x)$ (R) is called a gradient system if there exists a function $V(\vec x)$ such that $f(\vec x)=-\nabla V(\vec x).$ Show that if (R) is a gradient system then $$\frac{\partial f_i}{\partial x_i}-\frac{\partial f_j}{\partial x_j}=0.$$

Where do I begin with this?

Answer 1

Hint: Assume that the second partial derivative of $V$ are continuous, use the following fact: ${\partial^2\over{\partial_i\partial_j}}V={\partial^2\over{\partial_j\partial_i}}V$