>>11260411I'm don't like the other anon's answer, so have another one. I will use a physical system to explain it and give the intuition, because modeling physical systems is what all this math stuff was developed for anyway.
For a chemically stable closed system, the first law of thermodynamics says the any small change in internal energy of the system is equal to some small increment of heat plus a small increment of work. If all our processes are reversible, this means
Now, suppose our closed system is a pure gas inside a cylinder. Let us expand and compress that cylinder slow enough such that the temperature inside is always constant. Since internal energy is a function of temperature, this means . So,
What this means is that any increment of heat into the cylinder (the first term) is equal to the corresponding increment of work done by the gas (the second term above). Now, think about Green's theorem from calculus. This theorem says that the circulation of a function of two variables around a closed path which encloses a region is equal to zero if and only if that function is path independent. Equivalently,
If you make the proper substitutions of the variables from (*) (pressure, volume, entropy, temperature), into the equation above, you get one of Maxwell's relations, namely
which is an EXPERIMENTAL result. So the math predicts exactly what we see experimentally.
Tl;dr: M times a tiny change in x plus N times a tiny change in y equals zero, therefore dM/dy=dN/dx.
>>11260434ur d's suck
