How to differentiate a term with respect to metric

[the_doors]

Homework Statement

for example: $\frac{\partial(F^{ab}F_{ab})}{ \partial g^{ab}}$ where F_{ab} is electromagnetic tensor.
or $\frac{\partial N_{a}}{\partial g^{ab}}$ where $N_{a}(x^{b})$ is a vector field.

Homework Equations

The Attempt at a Solution

i saw people write $F^{ab}F_{ab}$ as $g^{ca}g^{db}F_{cd} F_{ab}$ and say we write the term in a way that exposes the dependence on the metric. but exactly what it means ?

 

[Orodruin]

You need to write your function in terms of the variables you have, in your case $g^{ab}$ and $F_{ab}$. The contravariant $F^{ab}$ is not independent of those and so will have non zero partial derivatives wrt both. Writing it in terms of your variables let's you differentiate easier.

As a similar example: Take $f(x,y) = xy$. The partial derivative $\partial_yf$ is simply $\partial_y (xy) = x$. This is essentially the same. Do not be confused by the fact that you use the same letter to denote the covariant and contravariant tensors.

 

[the_doors]

for example if $N_{\mu}(x^\nu)$ be unit time like dynamical four velocity, what's happening ? i don't know explicit form of $N_{\mu}$.

 

[the_doors]

ok we can write for first example:$\frac {\partial (g^{ca}g^{db}F_{cd}F_{ab})}{\partial g^{ef}}$ = $\frac{\partial g^{ca}}{\partial g^{ef}}g^{db}F_{cd}F_{ab}+\frac{\partial g^{db}}{\partial g^{ef}}g^{ca}F_{cd}F_{ab}+\frac{\partial F_{cd}}{\partial g^{ef}}g^{ca}g^{db}F_{ab}+\frac{\partial F_{ab}}{\partial g^{ef}}g^{ca}g^{db}F_{cd}$ . is it correct ?

$\frac{\partial g^{ca}}{\partial g^{ef}}=\delta^c_{e} \delta^a_{f}$ is it correct ?

what's happening for $\frac{\partial F_{cd}}{\partial g^{ef}}$ ?