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.

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