I am intrigued by the Jacobian method of deriving thermodynamic relations. For example, the four Maxwell relations for a single-phase, single-component system, whose fundamental equation is $dE = TdS - pdV$, can be obtained from the Jacobian relation $[TS] = [PV]$, using different choices of independent variables. The relation $[TS] = [PV]$ seems to be profound from a physical point of view, on par with fundamental equation itself.
To demonstrate the use of $[TS] = [PV]$, here is one of the four Maxwell relations, derived using Jacobians with respect independent variables $(T,V)$:
$$\frac{[TS]}{[TV]} = \frac{[PV]}{[TV]}$$ leads to $$(\frac{\partial S}{\partial V})_T = (\frac{\partial P}{\partial T})_V$$.
This suggests to me that, mathematically speaking, starting with the differential:
$$ dA = bdB + cdC \tag{1}$$
one should have:
$$ [bB] + [cC] = 0 \tag{2}$$.
That (2) should be a consequence of (1) seems obvious in light of thermodynamics, but I can't seem to derive it from any of the usual Jacobian identities. I strongly suspect the reason is that (2) requires (1) to be an exact differential, whereas Jacobian identities must also work under more general circumstances.
My question is simply: what is the most direct derivation of (2) from (1)?
