Consider the surface R², given by r(u, v) = (u, v). (a) Compute E, F and G. Verify that the matrix form of g is given by ( 1). du (b) Let v = Show that ||v||2 = g(v, v) = du² + dv². dv (c) Verify that ds = dA.
Consider the surface R², given by r(u, v) = (u, v). (a) Compute E, F and G. Verify that the matrix form of g is given by ( 1). du (b) Let v = Show that ||v||2 = g(v, v) = du² + dv². dv (c) Verify that ds = dA.
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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Pls explain in detail
![Problem 19*
Consider the surface R², given by r(u, v) = (u, v).
(a) Compute E, F and G. Verify that the matrix form of g is given by
[du]
(b) Let v = Show that |v||2 = g(v, v) = du² + dv².
dv
(c) Verify that ds = dA.
(1)
Therefore, with the metric as in part (a), on R², everything reduces to our usual working condition
i.e., Euclidean setting. The matrix given in part (a) is the Euclidean metric, which is an example of a
Riemannian metric on R².](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fca6b5de9-d666-4be4-bec5-372f49facd74%2F5f732c9a-728e-4f97-a24e-be797352e8f5%2F65pkiw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 19*
Consider the surface R², given by r(u, v) = (u, v).
(a) Compute E, F and G. Verify that the matrix form of g is given by
[du]
(b) Let v = Show that |v||2 = g(v, v) = du² + dv².
dv
(c) Verify that ds = dA.
(1)
Therefore, with the metric as in part (a), on R², everything reduces to our usual working condition
i.e., Euclidean setting. The matrix given in part (a) is the Euclidean metric, which is an example of a
Riemannian metric on R².
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