# Quick Calculation 3.1

A First Course in String Theory

.

Verify that $\displaystyle{\vec E}$, as given in (3.8), is invariant under the gauge transformation (3.10).

~~~

Eq. (3.8):

$\displaystyle{\vec E = - \frac{1}{c} \frac{\partial \vec A}{\partial t} - \nabla \Phi}$

Eq. (3.10):

\displaystyle{ \begin{aligned} \Phi' &= \Phi - \frac{1}{c} \frac{\partial \epsilon}{\partial t} \\ \vec A' &= \vec A + \nabla \epsilon \\ \end{aligned} }

.

\displaystyle{ \begin{aligned} \vec E' &= - \frac{1}{c} \frac{\partial \vec A'}{\partial t} - \nabla \Phi' \\ &= - \frac{1}{c} \frac{\partial \vec A}{\partial t} - \frac{1}{c} \frac{\partial}{\partial t} \left( \nabla \epsilon \right) - \nabla \Phi + \frac{1}{c} \nabla \frac{\partial \epsilon}{\partial t} \\ &= \vec E \\ \end{aligned} }

— Me@2022-04-01 03:34:28 PM

.

.