Differential notation help

[Castilla]

A question: is it possible (maybe handling differentials, I don't know) to have in the same formula these two expressions: $$\frac{dp}{dq}$$ and
$$\frac{dq}{dp}$$?

I think it is illogical. It would imply that at same time p is a function of q and q is a function of p. That seems nonsense to me. Am I allright?

Thanks for any answer.

 

[Hurkyl]

Isn't differential notation great?

There are several routes to happiness, but they all will give the same answer in the end (which is the reason such notation continues to be popular -- brevity is surprisingly important in mathematical notation).


One route to happiness is to have p be a function of q, and if the conditions of the inverse function theorem are satisfied, we can (locally) define a function $\hat{q}$ with the property that $\hat{q}(p(q)) = q$, and then we can say that $dq/dp$ really "means" $d\hat{q}/dp$.

 

[Castilla]

I have not here the paper where I found the formula that contained both
derivatives, but tomorrow I will get it and I will post it here with all its context. Then I will ask you, Hurkyl, if said formula is aceptable on grounds of the explanation that you kindly have posted. (Excuse my english).