# Definition
The magnetic vector potential in electromagnetism $\pmb{A}(pmb{r})$ is defined such that
$
\begin{align}
\nabla \times \pmb{A}(\pmb{r}) = \pmb{B}(\pmb{r}), \\
E(\pmb{x}) = -\nabla \phi(\pmb{r}) - \frac{\partial \pmb{A}(\pmb{r})}{t}
\end{align}
$
where $\pmb{B}(\pmb{r})$ is the [[magnetic field|magnetic field]], and $\pmb{E}(\pmb{r})$ is the [[electric field|electric field]]. $\phi(\pmb{r})$ is the [[electric potential|electric potential]] a [[scalar]] field.
More generally, a vector potential is a [[vector field]] whose [[Curl|curl]] is a given [[vector field]].
# Gauge Transformation
We will omit the argument $\pmb{r}$ from here on out.
The transformation
$
\pmb{A} \rightarrow \pmb{A} + \nabla \psi
$
for some [[scalar]] field $\psi$ leads to:
$
\begin{align}
\pmb{B} \rightarrow \pmb{B}, \\
\pmb{E} \rightarrow \pmb{E} - \nabla \frac{\partial \psi}{\partial t}, \\
\end{align}
$
Or, we can leave $\pmb{E}$ unchanged if we redefine the scalar potential as:
$
\phi \rightarrow \phi - \frac{\partial \psi}{\partial t}
$
## Coulomb Gauge
## Lorenz Gauge