Scalar Potential

A conservative Vector Field (for which the Curl $\nabla\times{\bf F}={\bf0}$) may be assigned a scalar potential

$\phi (x,y,z)-\phi(0,0,0) \equiv - \int_C {\bf F}\cdot {\bf ds}$
$= -\int_{(0,0,0)}^{(x,0,0)} F_1(t,0,0)\,dt + \int_{(x,0,0)}^{(x,y,0)} F_2(x,t,0)\,dt + \int_{(x,y,0)}^{x,y,z} F_3(x,y,t)\,dt,$

where $\int_C {\bf F}\cdot {\bf ds}$ is a Line Integral.

See also Potential Function, Vector Potential