Partial differentiation notation in thermodynamics In my thermodynamics notes, the definition of isothermal heat capacity at constant volume is
$$C_V := \frac T N \left(\frac{\partial S}{\partial T} \right)_V $$
and at constant pressure is
$$C_P := \frac T N \left(\frac{\partial S}{\partial T} \right)_P $$
where $T$ is temperature, $N$ is the number of molecules in the system and $S$ is entropy.
I don't fully understand the derivation symbols. To me, the partial derivative of entropy with respect to temperature would correspond to the "slope in the temperature direction", meaning that all other variables are considered to be "frozen". This would mean that the two partial derivatives above, and consequently $C_V$ and $C_P$, are one and the same object. But it is clearly not so.
Why is there a need to specify that we are keeping volume / pressure constant, when this is already implied in the definition of partial differentiation? How would one make this notation mathematically rigorous, i.e., what is the actual math behind this abbreviated notation? (Possibly taking differential forms into account?)

Whenever we take a partial derivative, we need to specify what we are keeping constant.  We commonly just use the partial derivative notation (say $\frac{\partial }{\partial x_1}$ in $\mathbb{R}^n$) to mean "keeping the other $n-1$ coordinates constant" and if that's what is meant, we usually don't need to say anything more.  But that is not the only meaning of the partial notation.  You can keep other things constant.  For example, in $\mathbb{R}^2$, you can ask, "what is $\frac{\partial f}{\partial x_1}$ keeping $(x_1+x_2)$ constant?".
In the definitions you have given, $C_v$ is the partial derivative w.r.t. T and indeed P is allowed to vary while V is held constant.  And in the case of $C_p$, V is allowed to vary as P is held constant.  
