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