Potential

A potential is a Scalar field V:A⊆R→RNV:A\subseteq \mathbb{R}\to \mathbb{R}^{N} associated with a Vector field F:A⊆RN→RN\mathbf{F}:A\subseteq \mathbb{R}^{N}\to \mathbb{R}^{N} such that its Gradient is the vector field with the sign changed:

F=−∇V\mathbf{F}=-\nabla V

The sign change is a convention and is designed to make equations nicer to work with, since many potentials are dependent on an inverse power of their argument (typically ∼1/r\sim 1/r, e.g. electric potential) which lead to a minus sign when derived. The additional minus in the definition cancel the other one out.

A potential is defined up to a constant, which means that given any constant CC, the function V~=V+C\tilde{V}=V+C is itself a potential of the same vector field. This is because the derivative of a constant is always zero:

F=−∇V~=−∇(V+C)=−∇V−∇C⏟0=−∇V\mathbf{F}=-\nabla \tilde{V}=-\nabla(V+C)=-\nabla V-\underbrace{ \nabla C }_{ 0 }=-\nabla V