Is it Logical to Have Both dp/dq and dq/dp in the Same Formula?

AI Thread Summary
The discussion revolves around the feasibility of including both dp/dq and dq/dp in the same formula, with one participant expressing skepticism about its logic. They argue that having both expressions implies a simultaneous dependency of p on q and q on p, which seems contradictory. Another participant suggests that if p is a function of q, the inverse function theorem allows for a local definition of a function that could justify the use of both derivatives. They plan to share a relevant paper to provide context for their argument. The conversation highlights the complexities and nuances of differential notation in mathematics.
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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.
 
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Isn't differential notation great? :rolleyes:

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).
 
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