# Homework Help: Proof of the existence of a scalar potential

1. Apr 6, 2009

### Mathmos6

1. The problem statement, all variables and given/known data
Hi there - I'm wondering about how you can actually show the existence of a scalar potential for an irrotational vector field E - if $\nabla \times E = 0$ everywhere, then how does one show there exists a scalar potential $\phi(x)$ such that $E=- \nabla \phi$?

3. The attempt at a solution
By Stokes' theorem we can see that $\int_C E dx = \int_S \nabla \times E dS = 0$ everywhere so our integral is path independent, but does path independence necessarily prove the existence of a scalar potential?

Thanks a lot, Mathmos6

2. Apr 6, 2009

### elect_eng

There is a well-known vector identity:

$$\bigtriangledown \times \bigtriangledown\phi=0$$

3. Apr 6, 2009

### Dick

Sure it does. Now pick a point A and define the potential at any point X as the path integral of E from A to X. It's well defined because of path independence.

4. Apr 6, 2009

### Mathmos6

Ah, fair enough - thanks Dick!