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2.A You are given the Euclidean space E³ with Cartesian coordinates Xi = {x, y, z) and standard line element ds2= dx²+dy²+dz². You are given the following surface: x = (1 + (v/2) cos(u/2)) cos(u), y = (1+ (v/2) cos(u/2)) sin(u) and z = (v/2) sin(u/2). (a) Compute the induced metric in coordinates Yi form = State the function f explicitly. {u, v}, i 1,2. It is of the = 1 ds² = f(u, v) du²+ = dv².

2.A You are given the Euclidean space E³ with Cartesian coordinates Xi = {x, y, z) and standard line element ds2= dx²+dy²+dz². You are given the following surface: x = (1 + (v/2) cos(u/2)) cos(u), y = (1+ (v/2) cos(u/2)) sin(u) and z = (v/2) sin(u/2). (a) Compute the induced metric in coordinates Yi form = State the function f explicitly. {u, v}, i 1,2. It is of the = 1 ds² = f(u, v) du²+ = dv².

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 You are given the Euclidean space E³ with Cartesian coordinates X¹ = {x, y, z} and standard line element ds² = dx²+dy²+dz². You are given the following surface: x = (1+ (v/2) cos(u/2)) cos(u), y = (1+ (v/2) cos(u/2)) sin(u) and z= = (v/2) sin(u/2). (a) Compute the induced metric in coordinates Y₁ = {u, v}, i = 1, 2. It is of the form 1 ds² = f(u, v) du² + = dv². State the function f explicitly. (b) Find 11, 12 and 1₁. The other Christoffel symbol components vanish.
Transcribed Image Text:2.A You are given the Euclidean space E³ with Cartesian coordinates X¹ = {x, y, z} and standard line element ds² = dx²+dy²+dz². You are given the following surface: x = (1+ (v/2) cos(u/2)) cos(u), y = (1+ (v/2) cos(u/2)) sin(u) and z= = (v/2) sin(u/2). (a) Compute the induced metric in coordinates Y₁ = {u, v}, i = 1, 2. It is of the form 1 ds² = f(u, v) du² + = dv². State the function f explicitly. (b) Find 11, 12 and 1₁. The other Christoffel symbol components vanish.
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