Hi! I'm wondering what mathematical principle let's me go from
$dE=TdS-PdV+\mu dN$
to
$dE=\left(\dfrac{\partial E}{\partial S}\right) _{V,N}dS+\left(\dfrac{\partial E}{\partial V}\right)_{S,N}dV+\left(\dfrac{\partial E}{\partial N}\right)_{S,V} dN$

I understand what to write to get from one to the other, but not what principle or principles make it acceptable.

Question

Hi! I'm wondering what mathematical principle let's me go from
$dE=TdS-PdV+\mu dN$
to
$dE=\left(\dfrac{\partial E}{\partial S}\right) _{V,N}dS+\left(\dfrac{\partial E}{\partial V}\right)_{S,N}dV+\left(\dfrac{\partial E}{\partial N}\right)_{S,V} dN$

I understand what to write to get from one to the other, but not what principle or principles make it acceptable.

theeric · Answer

Thanks in advance!

theeric · Answer

It's given that the second part is assuming that
$E=E(S,\ V,\ N)$

So I think that the second part comes just from the derivative of that.

theeric · Answer

So I'm not getting from one to the other... But they are related...

anonymous · Answer

Do you mind explaining what E, P,S are? :P

theeric · Answer

Well, I think it suffices that they're functions of time....
But I can't honestly say I'm 100% on that.

I don't think the physical meanings of the variables are important, but I'm sorry I didn't include them. Another important detail is that every derivative is with respect to time, because that is the style I've adopted from my professor and textbook.

The topic is thermodynamics in macroscopic physics.
Pertaining to any system in question (I think, maybe):
$E\equiv\rm{internal\ energy}\
P\equiv\rm{pressure}\
S\equiv\rm{entropy}\
N\equiv\rm{number\ of\ molecules}\\ \
T\equiv\rm{temperature}\
\mu\equiv\rm{chemical\ potential}$