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.
Hence a scalar function f exists so that
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.