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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
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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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.
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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