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Following David Tong's great lecture notes on QFT I've got struggling with the following steps he did. Can someone explain the steps between those three lines?

1.1.3 A final Example: Maxwell's Equations

We may derive Maxwell's equation in the vacuum from the Lagrangian, $$\mathcal{L}=-\frac{1}{2}(\partial_\mu A_\nu)(\partial^\mu A^\nu)+\frac{1}{2}(\partial_\mu A^\mu)^2 \tag{1.18}$$ Notice the funny minus signs! This is to ensure that the kinetic terms for $\mathbf{A}_i$ are positive using the Minkowski space metrtic (1.18), so $\mathcal{L}\sim\frac{1}{2}\dot A_i^2$. The Lagrangian (1.18) has no kinetic term $\dot A_0^2$ for $A_0$. We will see the consequences of this in Section 6. To see that Maxwell's equations indeed follow from (1.18), we compute $$\frac{\partial \mathcal{L}}{\partial(\partial_\mu A_\nu)}=-\partial^\mu A^\nu+(\partial_\rho A^\rho)\eta^{\mu\nu} \tag{1.19}$$ from which we may derive the equations of motion, $$\partial_\mu\left(\frac{\partial \mathcal{L}}{\partial(\partial_\mu A_\nu)}\right)=-\partial^2 A^\nu+\partial^\nu(\partial_\rho A^\rho) =-\partial_\mu(\partial^\mu A^\nu-\partial^\nu A^\mu)\equiv-\partial_\mu F^{\mu\nu} \tag{1.20}$$

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Uriel Zanzuri
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    Are you familiar with the Euler-Lagrange's equations in field theory? Also, you'd be better off writing down what you want to know using mathjax instead of just posting a screenshot of the text. – Davide Morgante Mar 31 '20 at 17:44
  • Hey Davide, I'm familiar with Euler Lagrange Equation. I couldn't understand the steps of tensor derivation between the first->second->three equation – Uriel Zanzuri Mar 31 '20 at 18:48

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