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We are to state and verify a necessary condition for the functional dependence of two differentiable functions $f(x, y)$ and $g(x, y)$ on an open subset $D \subset \mathbb{R}^2$. --- ### 1. Theorem (Necessary Condition for Functional Dependence) If two differentiable functions $f(x, y)$ and $g(x, y)$ are functionally dependent on an open set $D$, i.e., $$ F(f(x, y), g(x, y)) = 0 $$ for some differentiable $F(u,v)$, then the Jacobian determinant of $f$ and $g$ must vanish: ______ ______ ____ _________ _________ ____ ________ __________ ________.
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