The main point of this exercise is to use Green's Theorem to deduce a special case of the change of variable formula. Let U, V CR? be path connected open sets and let G:U → V be one-to-one and C2 such that the derivate DG(u) is invertible for all u E U. Let T CU be a regular region with piecewise smooth boundary, and let S = G(T). 2. (a) Prove that S is a regular region. [Hint: recall the proof that aS = G(@T)] Show that the Jacobian JG : U (u, v) → det(DG(u, v)) E R continuous. (b) [Hint: Don't work hard. Use algebraic properties of continuous functions.] (c) Deduce that Ja is either everywhere positive or evervwhere negative on U

The main point of this exercise is to use Green's Theorem to deduce a special case of the change of variable formula. Let U, V CR? be path connected open sets and let G:U → V be one-to-one and C2 such that the derivate DG(u) is invertible for all u E U. Let T CU be a regular region with piecewise smooth boundary, and let S = G(T). 2. (a) Prove that S is a regular region. [Hint: recall the proof that aS = G(@T)] Show that the Jacobian JG : U (u, v) → det(DG(u, v)) E R continuous. (b) [Hint: Don't work hard. Use algebraic properties of continuous functions.] (c) Deduce that Ja is either everywhere positive or evervwhere negative on U

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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