If a
vector function A satisfies
curl A = 0
then you can define a
scalar potential function V such that
grad V =
A
this makes things much easier to deal with, since you now have a (1-component) scalar function instead of an (arbitrarily-large-number-of-components) vector function. You lose accuracy to the degree of an arbitrary additive
constant, but that can be solved with the
magic of
boundary value problems.
A popular use of the potential function in
physics is the
electric potential, which is the potential function of the
electric field in magnetostatic problems. In problems where the
magnetic field is non-static, the potential formulation becomes more complicated.