Sure, I will try and simplify it. The squiggly things are meant to be partial derivatives
. Basically, we know that the internal energy changes if the temperature and/or the volume of the system changes, the term (δU/δT)
V is basically saying, how fast does U change when you make a small change in the temperature (at constant volume), the other term (δU/δV)
T is describing how much the internal energy changes when you make a small change in the volume (at constant temperature). Together, these describe the total internal energy change. For an ideal gas, the term (δU/δV)
T goes to zero, for reasons that I have explained in my previous post (i.e. the internal energy of an ideal gas does not change with volume). Hence you are left with the term (δU/δT)
V, which is simply the definition of the constant volume heat capacity. Integrating dU = Cv δT gives ΔU = C
vΔT.