I, too, spent a long time staring at expressions like

    “half-invert p(x, v) to get v(x, p) s.t. 

    p(x, v(x, q)) = q

    then the Legendre transform is 

    H(x, p) = p v(x, p) – L(x, v(x, p))”

And I did come to one of the same conclusions as this article, which is that if we're talking pure mathematics, these “thermodynamic” expressions like (∂L/∂v)_x, (∂L/∂p)_x are deeply easy to get confused about and in fact you should just say “the derivative of the function with respect to its first argument holding the other arguments constant” and therefore introduce different functions which compute the same value under different symbols, say

    Λ(x, p) = L(x, v(x, p))
    ∂₂Λ = ∂₂L ∂₂v

so that you're not scratching your head about “why is the derivative of L with respect to v showing up here, v is now a function isn't it?”

The formulation of first f derivatives as inverse functions is new to me but makes sense.

However, I do think that we do even worse with linear algebra. I believe I could walk up to any college senior in physics and they wouldn't know that “the determinant is the product of the eigenvalues,” but this should be as well-known as “the mitochondria are the powerhouse of the cell.” I think this is because we introduce a complicated way to calculate determinants and then we use determinants to calculate the eigenvalues?