Consider the unit sphere in R³. Suppose the parametrization is given by r(6,0) = (sin p cos , sin o sin , cos p), where and are same as described in the class. (a) Compute the Jacobian matrix, J of the parametrization. 0 (₁ 20). sin² Define g= helps us compute g easily in higher dimension, as we will see in the problem 19. (b) Verify that J¹ J is JTJ. This is another way of finding the metric g, which [do] (c) Let v = Verify that g(v, v) = do² +sin² p d0². This is called the round metric on the unit sphere. [ dᎾ (d) Verify that dS = sin o dA.
Consider the unit sphere in R³. Suppose the parametrization is given by r(6,0) = (sin p cos , sin o sin , cos p), where and are same as described in the class. (a) Compute the Jacobian matrix, J of the parametrization. 0 (₁ 20). sin² Define g= helps us compute g easily in higher dimension, as we will see in the problem 19. (b) Verify that J¹ J is JTJ. This is another way of finding the metric g, which [do] (c) Let v = Verify that g(v, v) = do² +sin² p d0². This is called the round metric on the unit sphere. [ dᎾ (d) Verify that dS = sin o 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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Transcribed Image Text:Problem 21*
Consider the unit sphere in R³. Suppose the parametrization is given by
r(0, 0)= (sin cos 0, sin ó sin ó, cos d),
where and are same as described in the class.
(a) Compute the Jacobian matrix, J of the parametrization.
0
(b) Verify that J¹ J is (1 sin²) Define g = J¹ J. This is another way of finding the metric g,
which
helps us compute g easily in higher dimension, as we will see in the problem 19.
(c) Let v =
Verify that g(v, v) = d² + sin² pd0². This is called the round metric on the unit sphere.
[ dᎾ
(d) Verify that dS = sin o dA.
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