The specific heat of a system is defined as
$$C_z = T \left( \frac{\partial S}{\partial T} \right)_{z=\text{const}}$$
Sometimes however, I find the same definition, but with total derivatives instead of partial derivatives:
$$C_z = T \left( \frac{d S}{d T} \right)_{z=\text{const}}$$
How can this be and what is the difference? Also, in class we calculated the specific heat of a superconductor from a given formula for the entropy. While we started off with the definition with partial derivatives, somewhere in the process the total derivatives started popping up out of nowhere. When a student asked why that is the teacher said something along the lines of "the partial derivative in the definition means the 'partial derivative in the thermodynamic sense'" and said that it's somehow equivalent to the total derivative, which I didn't understand.
So... what's the difference?
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