Derivation of the relativistic equation of energy conservation for a perfect fluid I'm currently attempting to struggle through the first chapter of Sean M. Carrol's spacetime and geometry. I'm a bit stuck, most likely because of not understanding the mathematical operation. Starting from the divergence of the energy-momentum tensor of a perfect fluid (1.116):
$\partial_\mu  T^{\mu\nu}= \partial_\mu (\rho+p)U^\mu U^\nu + (\rho+p)(U^\nu \partial_\mu U^\mu + U^\mu \partial_\mu U^\nu ) + \partial^\nu p$
I'm working under the assumption that $\partial^\nu p$ gives a vector, as opposed to $\partial_\nu p$ which gives a dual vector. 
Either way, the next step is to project it into pieces along and orthogonal to the four velocity field $U^\mu$, resulting in (1.118): $U_\nu \partial_\mu T^{\mu\nu} = - \partial_\mu (\rho U^\mu) - p \partial_\mu U^\mu$.
If I'm not mistaken, the idea is that any part of the equation that does not start with $U^\nu$ will drop out, because any such part is orthogonal to $U_\nu$. Parts that do begin with $U^\nu$ lose it, but gain a minus sign. If that were so, the result would be $-(\rho+p)\partial_\mu U^\mu + U_\nu \partial^\nu p$. I do not see how $\rho$ could be absorbed into the partial derivative. 
I'm a bit confused and surmise it is my unfamiliarity with Tensor calculations. 
My sincere thanks,
Daimonie
 A: 1- your "assumption" about the covariance of the derivatives of p is correct. This is just tensor calculus basics though, not a special assumption.
2-Projecting a vector/tensor along different directions is a practice widely used in math and physics. For example you probably encountered this first when you decomposed the gravitational force on a block on inclined plane in Physics 101. Here you use a vector $U_{\nu}$ and a projector on a orthogonal plane to $U$.
3- You are forgetting the first part of RHS of eq. 1.116. Proceed step by step.
Let's multiply the RHS of 1.116 by $U_{\nu}$
$U_{\nu} RHS = U_{\nu}U^\mu U^\nu \partial_\mu (\rho+p)+ (\rho+p)(U_{\nu}U^\nu \partial_\mu U^\mu + U_{\nu}U^\mu \partial_\mu U^\nu ) + U_{\nu}\partial^\nu p$
Using $U_{\nu} U^\nu =-1$
$-U^\mu\partial_\mu (\rho+p)+ (\rho+p)(-\partial_\mu U^\mu + U_{\nu}U^\mu \partial_\mu U^\nu ) + U_{\nu}\partial^\nu p$
Now remember that  $ U_{\nu} \partial_\mu U^\nu =0 $ (Eq. 1.117 in Carrol) as a consequence of the constancy of $U_{\nu} U^\nu =-1$. So we further simplify
$-U^\mu\partial_\mu (\rho+p)+ (\rho+p)(-\partial_\mu U^\mu + 0 ) + U_{\nu}\partial^\nu p$
Expand round brackets
$-U^\mu\partial_\mu \rho -U^\mu\partial_\mu  p - \rho\partial_\mu U^\mu-p\partial_\mu U^\mu + U_{\nu}\partial^\nu p$
Now, the second term $-U^\mu\partial_\mu p $  can be rewritten as $-U^\nu\partial_\nu p $ since the mu's are dummy, and you can also turn the indices up/down to get $U_{\nu}\partial^\nu p$ and cancel it with the last term. So you get
$-U^\mu\partial_\mu \rho  - \rho\partial_\mu U^\mu-p\partial_\mu U^\mu $
Now combine the first 2 terms (anti-Leibnitz rule) and get the RHS of Eq. 1.118 in Carrol
$ - \partial_\mu( \rho U^\mu)-p\partial_\mu U^\mu $
I suggest if you are new to GR and tensors:
1- get fit doing some "index gymnastics"
2- check out other books (my favorite is Gravitation aka MTW)
3- get access/buy, use and learn Wolfram Mathematica (home or student versions) perhaps thru your University. You can then get a free package called xAct and its free GUI called xPrint (which I wrote). Then...you will do all this and much much more in a flash. Mathematica is an essential physicist's tool in my opinion.
