Quick Calculation 3.1

A First Course in String Theory

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

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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}  }

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\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

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2022.04.01 Friday (c) All rights reserved by ACHK