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Consider Euclidean space in D = 2. Show by direct computation that the Riemann or,

Consider Euclidean space in D = 2. Show by direct computation that the Riemann or,

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. Consider Euclidean space in D = 2. Show by direct computation that the Riemann curvature tensor, Horvp-roro HP RA рuz = ангир - ,Г³ up +r vanishes in: a) Cartesian coordinates, such that ds² = dx² + dy². b) polar coordinates, such that ds² = dr² + r²d0². You can use the fact that the only non-vanishing Christoffel symbols are I 00= -r, since these were computed in a recent example class. [Hint: Notice that the Riemann tensor has a single independent component in D = 2, due to its symmetries.] Tºrº = r9 Or =
Transcribed Image Text:4. Consider Euclidean space in D = 2. Show by direct computation that the Riemann curvature tensor, Horvp-roro HP RA рuz = ангир - ,Г³ up +r vanishes in: a) Cartesian coordinates, such that ds² = dx² + dy². b) polar coordinates, such that ds² = dr² + r²d0². You can use the fact that the only non-vanishing Christoffel symbols are I 00= -r, since these were computed in a recent example class. [Hint: Notice that the Riemann tensor has a single independent component in D = 2, due to its symmetries.] Tºrº = r9 Or =
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