## Finding the Scalar Potential Function for an Irrotational Vector Field

If a vector fieldhas zero curl everywhere, so thatat every point, then for some functioncalled the scalar potential.

Givenwe can findat least to an additive constant, by forming and solving differential equations.

Example:

Hence a scalar function f exists so that

Then

Integrating the first of these gives

Integrating the second of these gives

Integrating the third of these gives

andare arbitrary functions and sinceis a functions ofonly,is a function ofonly andis a function ofonly,andare constant functions.

Hence