Electromagnetic field equations of motion

[smallgirl]

1. I'm not quite sure how the laplacian acts on this integral
2. \frac{\delta S}{\delta A_{\mu}}=\int\frac{\delta}{\delta A_{\mu}}(\frac{1}{4}F_{\rho\sigma}\frac{\triangle}{M^{2}}F^{\rho\sigma})
3. I know I have to split the integral into three integrals for x y and z, but I'm not sure if
a) I should write out F in full,
b) If the laplacian only acts on the second F or both, and if so how for each instance

I have got this anyways =\int\frac{\delta}{\delta A_{\mu}}\frac{1}{4M^{2}}(\partial_{\rho}A_{\sigma}-\partial_{\sigma}A_{\rho})\frac{\partial^{2}}{\partial x^{2}}(\partial^{\rho}A^{\sigma}-\partial^{\sigma}A^{\rho})+\int\frac{\delta}{\delta A_{\mu}}\frac{1}{4M^{2}}(\partial_{\rho}A_{\sigma}-\partial_{\sigma}A_{\rho})\frac{\partial^{2}}{\partial y^{2}}(\partial^{\rho}A^{\sigma}-\partial^{\sigma}A^{\rho})+\int\frac{\delta}{\delta A_{\mu}}\frac{1}{4M^{2}}(\partial_{\rho}A_{\sigma}-\partial_{\sigma}A_{\rho})\frac{\partial^{2}}{\partial z^{2}}(\partial^{\rho}A^{\sigma}-\partial^{\sigma}A^{\rho})

 

[member 553083]

4. The answer is \frac{\delta S}{\delta A_{\mu}}=\int\frac{1}{M^{2}}\left[\partial^{\alpha}A_{\sigma}\frac{\partial^{2}}{\partial x^{2}}(\partial_{\alpha}A^{\sigma}-\partial^{\sigma}A_{\alpha})+\partial^{\alpha}A_{\sigma}\frac{\partial^{2}}{\partial y^{2}}(\partial_{\alpha}A^{\sigma}-\partial^{\sigma}A_{\alpha})+\partial^{\alpha}A_{\sigma}\frac{\partial^{2}}{\partial z^{2}}(\partial_{\alpha}A^{\sigma}-\partial^{\sigma}A_{\alpha})\right]