Total and partial derivatives of $F(T, X(Y, Z))$ respect to $T$? So, yeah, haven't been to vector analysis just yet, but for now I'd need to make sure I understand the definition of total derivatives (and partial derivatives). 
The question is simple. I wish to take $\frac{dF}{dT}$ when $F = F(T, X(Y,Z))$ so that Y and Z might depend on T.
And just to check that I understand the rule correctly, what is $\frac{\partial F}{\partial T}$? Do I always have to specify which variable I keep constant when taking the partial derivative - I somehow undestood that when taking a partial derivative it didn't matter, as you only take derivatives respect to explicit dependence, and that's it. 
But then I've seen the use of $(\frac{\partial F}{\partial T})_{X}$, say. What does that mean? 
It would be awesome if someone could clarify =)
 A: Just use chain rule over and over:
$$
\frac {dF}{dT} = \frac {\partial F}{\partial T} + \frac {\partial F}{\partial X} \frac {\partial X}{\partial T} = \frac {\partial F}{\partial T} + \frac {\partial F}{\partial X} \left( \frac {\partial X}{\partial Y} \frac {\partial Y}{\partial T} +  \frac {\partial X}{\partial Z} \frac {\partial Z}{\partial T}\right ) \tag {*}
$$
Best way to check it is direct calculations. So, for example, let's say you have a function $F(T, X) = T^2 + \sin X$, but $X(Y,Z) = Y^5 + \frac 1Z$, $Y = \log T$ and $Z = \exp T$, so $$
F = T^2 + \sin \left( \log^5 T - \exp T\right)
$$
Total derivation means that you find derivative of $F$ with respect to $T$ when you reduce all other arguments to $T$ only, like I did above. So
$$
\frac {dF}{dT} = 2T + \cos \left( \log^5 T - \exp T\right) \left(\frac{5 \log^4 T}T - \exp T\right )
$$
On the other hand $\frac {\partial F}{\partial T}$ means partial derivative, i.e. when all other arguments are considered constant. So, for the example above
$$
\frac {\partial F}{\partial T} = 2T
$$
In your case, you have a function of two variables only (initially), so
$$
\left (\frac {\partial F}{\partial T} \right)_X \equiv \frac {\partial F}{\partial T}
$$
