##### Potentials
c.m.vf.pot

If the curl of a vector field \({\bf F}\) vanishes, then \({\bf F}\) can be written as the gradient of a scalar potential \(V\):

\begin{equation} {\boldsymbol \nabla} \times {\bf F} = 0 \Longleftrightarrow {\bf F} = -{\boldsymbol \nabla} V \label{Gr(1.103)} \end{equation}

###### Theorem 1: Curl-less (irrotational) fields

The following conditions are equivalent:

- (a) \({\boldsymbol \nabla} \times {\bf F} = 0\) everywhere.
- (b) \(\int_{\bf a}^{\bf b} {\bf F} \cdot d{\bf l}\) is independent of path for any given end points.
- (c) \(\oint {\bf F} \cdot d{\bf l} = 0\) for any closed loop.
- (d) \({\bf F}\) is the gradient of some scalar field, \({\bf F} = -{\boldsymbol \nabla}V\).

If the divergence of a vector field \({\bf F}\) vanishes, then \({\bf F}\) can be expressed as the curl of a {\bf vector potential} \({\bf A}\).

###### Theorem 2: Divergence-less (solenoidal) fields

The following conditions are equivalent:

- (a) \({\boldsymbol \nabla} \cdot {\bf F} = 0\) everywhere.
- (b) \(\int {\bf F} \cdot d{\bf a}\) is independent of surface, for any given boundary line.
- (c) \(\oint {\bf F} \cdot d{\bf a} = 0\) for any closed surface.
- (d) \({\bf F}\) is the curl of some vector, \({\bf F} = {\boldsymbol \nabla} \times {\bf A}\).

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42