Vector potential

A vector potential is a Vector field A:S⊆RN→RN\mathbf{A}:S\subseteq \mathbb{R}^{N}\to \mathbb{R}^{N} associated with another vector field F:S⊆RN→RN\mathbf{F}:S\subseteq \mathbb{R}^{N}\to \mathbb{R}^{N} such that its Curl is the vector field:

∇×A=F\nabla\times\mathbf{A}=\mathbf{F}

The vector potential is defined up to the gradient of a Scalar field, which means that given any scalar field SS, the function A~=A+∇S\tilde{\mathbf{A}}=\mathbf{A}+\nabla S is itself the vector potential of the same field. This is because the curl of a Gradient is always zero:

F=∇×A~=∇×(A+∇S)=∇×A+∇×(∇S)⏟0=∇×A\mathbf{F}=\nabla\times \tilde{\mathbf{A}}=\nabla \times(\mathbf{A}+\nabla S)=\nabla\times\mathbf{A}+\underbrace{ \nabla \times(\nabla S) }_{ 0 }=\nabla\times\mathbf{A}