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
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feaeaf68c-12bd-4432-afa6-6f1bd2c82cdd%2F6eb2d8e5-cbf1-473f-8748-610c96e3a0e3%2Fbcdrbd_processed.png&w=3840&q=75)
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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