Calculation mistake some place in finding stress-energy tensor If the Lagrangian in Maxwell's theory is $$L= R- \frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$ 
I want to find $T_{\mu\nu} $
The procedure is that I vary the action:
$$\delta S = -1/2 \int{d^4x \sqrt{g}(g_{\mu\nu}\delta g^{\mu\nu}L-2\delta L)}$$
I calculated $$\delta L = 2(\delta g^{\mu\nu})F_{\nu}\, ^{\lambda} F_{\mu\lambda}$$
Then I was substituting things back by ignoring the R in the lagrangian above for a reason I don't recognize but I was aiming to reach the final answer which was $T_{\mu\nu}$  - I looked up in google to find it equal to $$T_{\mu\nu}= F_{\mu\lambda} F^\lambda_\nu - 1/4g_{\mu\nu}F_{\rho\sigma}F^{\rho\sigma}$$
As I substituted my $\delta L$, and $L$ and again  ($L$ being deprived  of R) I got:
$$\delta S = \int{ d^4x \sqrt{g} \delta g^{\mu\nu}(-1/4 g_{\mu\nu}F^{\mu\nu}F_{\mu\nu} + g_{\mu\nu}F _\nu \, ^\lambda F_{\mu\lambda})}$$Thus because of the relation $\delta S = \int{ d^4x \sqrt{g} \delta g^{\mu\nu}(T_{\mu\nu})}$ I knew by identification that my Tensor is $$-1/4 g_{\mu\nu}F^{\mu\nu}F_{\mu\nu} + g_{\mu\nu}F_\nu \, ^\lambda F_{\mu\lambda}.$$
Comparison with the one I found on the web I don't know how to get rid of the $g_{\mu\nu} $in the second term in my tensor and I do not know why my  F indices in the first term are the same as those of the metric g unlike the tensor I looked up in google.
May you please help me find my mistake?
 A: $g_{\mu \nu}F^{\mu \nu}F_{\mu \nu}$ does not make any sense (or at the very least is horrible notation).  Instead, I suspect you mean $g_{\mu \nu}F^{\rho \sigma}F_{\rho \sigma}$.  With this, it seems as if your answer would agree with Google's.
I believe you made this mistake because you had the expression
$$
g_{\mu \nu}L,
$$
and the second term of $L$ is $-\frac{1}{4}F_{\mu \nu}F^{\mu \nu}$.  Of course, before you substitute, you should change the name of $\mu$ and $\nu$.  For example, we would not write $x^3\cdot \int _2^3\mathrm{d}x\, \exp (x)$, instead we would write $x^3\cdot \int _2^3\mathrm{d}t\, \exp (t)$ or something similar.
A: You duplicated naming the dummy indices. There are two kinds of indices
1) Free index : the index which is not contract to any others.For these indices You must fix their name and all must be different.
2) Dummy index : the index which is  contract to the another one. For this type of index, you can change their name freely but fix the pairing and the name of the indices in different pair need to be different.
