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-)

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 https://arxiv.org/pdf/1811.04227 Or this Physics SE discussion: https://physics.stackexchange.com/questions/388318/how-exactly-is-the-formalism-of-thermodynamics-based-on-contact-geometry https://physics.stackexchange.com/questions/388318/how-exact...