Does this notation for partial derivatives work?

1. Oct 28, 2009

Gerenuk

I noticed that for partial calculus equations I can use Jacobians
$$\left(\frac{\partial A}{\partial B}\right)_C=\frac{\partial(A,C)}{\partial(A,B)}$$
You immediately get the triple product rule
(http://en.wikipedia.org/wiki/Triple_product_rule)
$$\frac{\partial(A,C)}{\partial(A,B)}=\frac{\partial(A,C)}{\partial(A,B)}\frac{\partial(B,C)}{\partial(B,C)}=-\frac{\partial(A,C)}{\partial(B,C)}\frac{\partial(B,C)}{\partial(B,A)}$$
and of course the other rule
$$\frac{\partial(A,C)}{\partial(A,B)}=\frac{\partial(A,C)}{\partial(A,B)}\frac{\partial(A,D)}{\partial(A,D)}=\frac{\frac{\partial(A,C)}{\partial(A,D)}}{\frac{\partial(A,B)}{\partial(A,D)}}$$

Now I was surprised to see that even the pseudo-equation for the chain rule
$$\partial(A,B)=\frac{\partial(A,D)\partial(E,B)-\partial(A,E)\partial(D,B)}{\partial(E,D)}$$
works well even if I treat all terms as a real variable.

My question is: Does this algebra for partial derivatives always work despite crazy manipulations and cancellations?