Discussion
Consider the following statements about partial derivatives. 1. The mixed derivatives commute whenever they are continuous 2. Euler's theorem weights each first derivative by its own variable 3. A vanishing Jacobian means the functions are independent
Answer: B
Clairaut's theorem gives the first, and Euler's theorem \( x f_{x} + y f_{y} = n f \) the second. The third reverses the truth: a vanishing Jacobian marks functional *dependence*, a relation \( \phi(u,v) = 0 \) tying the two together, and independence is what a non-zero Jacobian certifies
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