This is a problem from my lecture on analytical mechanics:
1) Consider a function $U$ of two variables $$U = U(S,V) \quad \text{with } \quad dU = TdS-PdV.$$ Construct the Legendre transform of $U$ which uses $T$ as independent variable and calculate its derivative with respect to $T$ and $V$.
I find this problem very confusing for 2 reasons:
I was under the impression that we could only calculate the Legendre transform of a real valued convex function of one variable;
$T$ is not a variable of $U$ so why does the problem state that $T$ is and independent variable of $U$ ?
Could someone clarify what I'm supposed to be doing in this exercise ?