3.2 Maxwell equations in four dimensions, 2

A First Course in String Theory

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(b) Show explicitly that the Maxwell equations with sources emerge from (3.34).

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Eq. (3.34):

\displaystyle{\begin{aligned}    \frac{\partial F^{\mu \nu}}{\partial x^\nu} &= \frac{1}{c} j^\mu \\       \end{aligned}}

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\displaystyle{\begin{aligned}      \frac{\partial F^{\mu 0}}{\partial x^0} + \frac{\partial F^{\mu 1}}{\partial x^1}   + \frac{\partial F^{\mu 2}}{\partial x^2} + \frac{\partial F^{\mu 3}}{\partial x^3}   &= \frac{1}{c} j^\mu \\ \\    \end{aligned}}

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\displaystyle{\begin{aligned}      \frac{\partial F^{0 0}}{\partial x^0} + \frac{\partial F^{0 1}}{\partial x^1}   + \frac{\partial F^{0 2}}{\partial x^2} + \frac{\partial F^{0 3}}{\partial x^3}   &= \frac{1}{c} j^0 \\     \frac{\partial E_x}{\partial x^1} + \frac{\partial E_y}{\partial x^2} + \frac{\partial E_z}{\partial x^3}   &= \rho \\     \nabla \cdot \mathbf{E} &= \rho \\ \\    \end{aligned}}

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\displaystyle{\begin{aligned}     \frac{\partial F^{1 0}}{\partial x^0} + \frac{\partial F^{1 1}}{\partial x^1}   + \frac{\partial F^{1 2}}{\partial x^2} + \frac{\partial F^{1 3}}{\partial x^3}   &= \frac{1}{c} j^1 \\     - \frac{\partial E_x}{\partial x^0} + \frac{\partial 0}{\partial x^1}   + \frac{\partial B_z}{\partial x^2} - \frac{\partial B_y}{\partial x^3}   &= \frac{1}{c} j^1 \\     \frac{\partial B_z}{\partial y} - \frac{\partial B_y}{\partial z}   &= \frac{1}{c} j^1 + \frac{\partial E_x}{\partial x^0} \\         \end{aligned}}

— Me@2022.10.02 11:49:59 AM

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2022.10.02 Sunday (c) All rights reserved by ACHK