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2. (a) For 0

2. (a) For 0

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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2. (a) For 0<u< 2 and 0 < v< 2л, let σ(u, v) = (u, v) be the part of xy-plane. Determine the Gauss curvature, say Kp, of the given surface P. (b) For 0<u<2 and 0 < v<2π, let 8(u, v) be the part of a cylinder. Write down the parametrization of 8(u, v) and determine the Gauss curvature, say Kc, of the given cylinder C. (c) Write down the definition of local isometry. (d) Show that the given plane P and the given cylinder C are local isometric to each other. (e) Show that Theorem 10.2.1 (Gauss' Theorema Egregium) holds for the given plane P and the given cylinder C.
Transcribed Image Text:2. (a) For 0<u< 2 and 0 < v< 2л, let σ(u, v) = (u, v) be the part of xy-plane. Determine the Gauss curvature, say Kp, of the given surface P. (b) For 0<u<2 and 0 < v<2π, let 8(u, v) be the part of a cylinder. Write down the parametrization of 8(u, v) and determine the Gauss curvature, say Kc, of the given cylinder C. (c) Write down the definition of local isometry. (d) Show that the given plane P and the given cylinder C are local isometric to each other. (e) Show that Theorem 10.2.1 (Gauss' Theorema Egregium) holds for the given plane P and the given cylinder C.
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