Partial Differentiation in Statistical Mechanics

Question

I am damn struggling with basics in here. I know that $U=U(N,V,T)$ and $z=z(N,V,T)$ so that $N=N(z,V,T)$. Now, I want to do partial differentiation using chain rule involving three variables so that I need to prove :$$\left (\frac{\partial U}{\partial T} \right )_{z,V} =\left ( \frac{\partial U}{\partial T} \right )_{N,V}+\left ( \frac{\partial U}{\partial N} \right )_{V,T}\left ( \frac{\partial N}{\partial T}\right )_{z,V} $$ I am actually gettting stuck into the partial differentiation involving three variables

Answer 1

1. Consider U to be a function N,V,T. Calculate its differential, dU. This will be expression 1. The expression will contain dN, dV and dT.
2. Consider N to be a function of z,V,T. Calculate its differential, dN. Substitute this expression for dN back into expression 1. You should get an expression for dU in terms of dz, dV and dT. This is expression 2.
3. Consider U to be a function of z,V,T. Calculate its differential, dU in terms of dz, dV, dT. This is expression 3.
4. Argue that the coefficient of dT in expression 2 must be equal to the coefficient of dT in expression 3.
5. Be sure to use the subscripts on your partial derivatives to indicate explicitly what variables are being held constant.