# Question: If is continuous

If is continuous, show that

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Use Fubini's Theorem to give an easy proof that D1, 2f = D2, 1f if these are continuous.Let g1, g2: R2 → R be continuously differentiable and suppose D1 g2= D2 R1.. As in Problem 2-21, letFind a counter-example to Theorem 5-2 if condition (3) is omitted. Following the hint, consider f: (- 2π, 2π) →R2 defined bySuppose C is a collection of coordinate systems for M such that (1) For each x Є M there is f Є C which is a coordinate system around ; (2) if f, g Є C, then det (f -1 0 g) 2 > 0. Show that there is a ...a. Show that Theorem 5-5 is false if M is not required to be compact. b. Show that Theorem 5-5 holds for noncom-pact M provided that w vanishes outside of a compact subset of M.Post your question