# Differential notation help

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?

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Hurkyl
Staff Emeritus
Gold Member
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" $d\hat{q}/dp$.

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).