2. 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, VC R2 be path connected open sets and let G: UV be one-to-one and C2 such that the derivate DG(u) is invertible for all u EU. Let TCU be a regular region with piecewise smooth boundary, and let S = G(T). (c) (d) Prove that S is a regular region. [Hint: recall the proof that JS = G(OT)] Show that the Jacobian JGU (u, v) → det (DG(u, v)) E R continuous. [Hint: Don't work hard. Use algebraic properties of continuous functions.] Deduce that JG is either everywhere positive or everywhere negative on U. If JG (u, v) > 0 for all (u, v) E U, convert the formula Area(S) = -fas ydx into an integral over OT using a change of variable, and then apply Green's Theorem to show that Area(S) = ff det (DG(u, v))dA. (e) If JG (u, v) < 0 for all (u, v) EU, use a similar argument to show that Area (S) - ff det (DG(u, v))dA. Where does the minus sign come from? =
2. 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, VC R2 be path connected open sets and let G: UV be one-to-one and C2 such that the derivate DG(u) is invertible for all u EU. Let TCU be a regular region with piecewise smooth boundary, and let S = G(T). (c) (d) Prove that S is a regular region. [Hint: recall the proof that JS = G(OT)] Show that the Jacobian JGU (u, v) → det (DG(u, v)) E R continuous. [Hint: Don't work hard. Use algebraic properties of continuous functions.] Deduce that JG is either everywhere positive or everywhere negative on U. If JG (u, v) > 0 for all (u, v) E U, convert the formula Area(S) = -fas ydx into an integral over OT using a change of variable, and then apply Green's Theorem to show that Area(S) = ff det (DG(u, v))dA. (e) If JG (u, v) < 0 for all (u, v) EU, use a similar argument to show that Area (S) - ff det (DG(u, v))dA. Where does the minus sign come from? =
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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