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(a) Let g(x, y): R² → R² by given by g(x, y) (2ye2, re). Show that g is a bijection between a neighbourhood of (0, 1) and a neighborhood of (2,0) and compute D(g-¹)(2,0). (b) Let ƒ : R² → R² be given by f(x, y) = (x² - y², 2xy). Let A = {(x, y): x>0} CR². Show f is a bijection between A and f(A) and compute D(f-¹)(0, 1). (Hint: if a = b then ||a|| = ||b||)

(a) Let g(x, y): R² → R² by given by g(x, y) (2ye2, re). Show that g is a bijection between a neighbourhood of (0, 1) and a neighborhood of (2,0) and compute D(g-¹)(2,0). (b) Let ƒ : R² → R² be given by f(x, y) = (x² - y², 2xy). Let A = {(x, y): x>0} CR². Show f is a bijection between A and f(A) and compute D(f-¹)(0, 1). (Hint: if a = b then ||a|| = ||b||)

Advanced Engineering Mathematics
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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4. (a) Let g(x, y): R² R² by given by g(x, y) = (2ye²,re). Show that g is a bijection between a neighbourhood of (0, 1) and a neighborhood of (2,0) and compute D(g¯¹)(2,0). (b) Let ƒ : R² → R² be given by f(x, y) = (x² − y², 2xy). Let A = {(x, y): x>0} CR2. Show f is a bijection between A and f(A) and compute D(ƒ-¹)(0,1). (Hint: if a = b then ||a|| = ||b||)
Transcribed Image Text:4. (a) Let g(x, y): R² R² by given by g(x, y) = (2ye²,re). Show that g is a bijection between a neighbourhood of (0, 1) and a neighborhood of (2,0) and compute D(g¯¹)(2,0). (b) Let ƒ : R² → R² be given by f(x, y) = (x² − y², 2xy). Let A = {(x, y): x>0} CR2. Show f is a bijection between A and f(A) and compute D(ƒ-¹)(0,1). (Hint: if a = b then ||a|| = ||b||)
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