I am trying to understand the the following derivation:
[itex] Cv = (qv/ΔT) = (ΔU/ΔT) \\
Cv = (∂U/∂T)v \\
dU = CvdT [/itex]
The Attempt at a Solution
So here is what I understand so far. I understand that heat transfer q and temperature T are related by a direct proportionality constant C. I also understand that change in internal energy is equal to heat transfer. What I don't understand is how the book goes from change in deltaU and deltaT to partial derivative of U with respect to T while volume is constant (i.e. how did they go from line 1 to line 2).
(For small deltaT U is a linear function of T, deltaU = CvdeltaT)
I phrased my question as "Relationship between deltax, partialX and dx" because I suspect my confusion lies in not understanding what the relationship is between these different symbols.
Thank you very much