1. Let T(u, v) = (u²-v², 2uv). Let D(u, v) be the region in the uv-plane with u² + v² ≤ 1, u ≥ 0 and v≥ 0. (a) Sketch D(u, v). (b) Sketch T(D(u, v)). Hints: i. Where do (0, v), (u,0) and u²+ v² = 1 map to? ii. Relate (u² + ²)² and (u² - v²)². (c) Calculate the Jacobian of T.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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1. Let T(u, v): =
v ≥ 0.
(u²v², 2uv). Let D(u, v) be the region in the uv-plane with u² + v² ≤ 1, u ≥ 0 and
(a) Sketch D(u, v).
(b) Sketch T(D(u, v)). Hints:
i. Where do (0, v), (u,0) and u² + v² = 1 map to?
ii. Relate (u²+v²)² and (u² — v²)².
(c) Calculate the Jacobian of T.
Transcribed Image Text:1. Let T(u, v): = v ≥ 0. (u²v², 2uv). Let D(u, v) be the region in the uv-plane with u² + v² ≤ 1, u ≥ 0 and (a) Sketch D(u, v). (b) Sketch T(D(u, v)). Hints: i. Where do (0, v), (u,0) and u² + v² = 1 map to? ii. Relate (u²+v²)² and (u² — v²)². (c) Calculate the Jacobian of T.
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