Consider Euclidean space in D = 2. Show by direct computation that the Riemann or,
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
=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F93fc8a26-029d-4535-b354-88f8685088a7%2Fa5f8266e-086a-4034-87f1-3a3b2f8e5a2b%2Fh0flwab_processed.png&w=3840&q=75)
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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