[bubblyworld]

I assume you're talking about thermodynamics - this comes down to a slight abuse of notation. For an ideal gas, say, you can express various state functions like the internal energy in various different ways. You can do it in terms of pressure P and volume V to get U ~ PV, for instance.

Or you could do it in terms of temperature T and pressure, for instance, to obtain U ~ T (in this case there's no dependence on pressure).

The ideal gas laws let you transform between these choices. But the point is that the same physical quantity, U, has multiple mathematical functions underlying it - depending on which pair you choose to describe it with!

To disambiguate this physicists write stuff like (dU/dP)_T, which means "partial derivative of U wrt P, where we use the expression for U in terms of P and T". Note that this is not the same as (dU/dP)_V, despite the fact that it superficially looks like the same derivative! The former is 0 and the latter is ~V, which you can compute from the expressions I gave above.

The mistake is thinking that U is a single function of many independent variables P, T, S, V, etc. Actually these variables all depend on each other! So there are many possible functions corresponding to U in a formal sense, which is something people gloss over because U is a single physical quantity and it's convenient to use a single letter to denote it.

Maybe it would make more sense to use notation like U(T, P) and U(P, V) to make it clear that these are different functions, if you wanted to be super explicit.

[AnimalMuppet]

> The mistake is thinking that U is a single function of many independent variables P, T, S, V, etc. Actually these variables all depend on each other!

So, in vector space terms, we have different bases for describing U in, but not that many independent variables.

If U is a function of x and y, but x and y are not orthogonal, then I can't treat dU/dx and dU/dy as independent, even for partial derivatives, because x and y aren't really independent.

[Y_Y]

You're not, in general, just working in a vector space but on a manifold whose coordinates are your extensive variables. It's only linear locally, in the (co-)tangent space where you're doing calculus.

See e.g. https://arxiv.org/pdf/1811.04227

Or this Physics SE discussion: https://physics.stackexchange.com/questions/388318/how-exact...

[bubblyworld]

Yeah, I think this is along the right lines - in the vector space analogy it's like we have a bunch of vectors we can measure (P, T, S, V, etc) but due to the constraints we're actually working in a 2 dimensional space. So we could form a basis from many different choices of vectors, and our coefficients would change accordingly.

As the other commenter said, you can make this analogy rigorous by looking at manifolds (differential geometry). They're a little bit like the non-linear version of a vector space. In this case the set of physically valid values for P, T, S, V forms a two-dimensional surface due to the ideal gas laws, and you can derive local coordinate charts for the surface using any (non-degenerate) pair of these variables.

[moi2388]

Thank you, these comments were really enlightening for me :)

[slooonz]

Yes, exactly the explanation I looked for, thank you !