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The function x(u, v) (R cos(u) cos(v), R sin(u) cos(v), R sin(v)) is a local parametrization of the sphere S of radius R> 0 centered at the origin. Find the components E, F, G of the first fundamental form of S and then use the systems of equations (²₂E+r²₁ F Eu T₁F+r₁G = F₁-E to compute the Christoffel Symbols T₁₂E+T²₂G = E₁ ₁₂F+1₂G = Gu F₁, F₁, F₂ = 11: 111 [₂2E+²₂F = F₂ - G₁ TF+G = ₂₁, 22 = 2₁, 22, and I'22. 21¹ 21¹

The function x(u, v) (R cos(u) cos(v), R sin(u) cos(v), R sin(v)) is a local parametrization of the sphere S of radius R> 0 centered at the origin. Find the components E, F, G of the first fundamental form of S and then use the systems of equations (²₂E+r²₁ F Eu T₁F+r₁G = F₁-E to compute the Christoffel Symbols T₁₂E+T²₂G = E₁ ₁₂F+1₂G = Gu F₁, F₁, F₂ = 11: 111 [₂2E+²₂F = F₂ - G₁ TF+G = ₂₁, 22 = 2₁, 22, and I'22. 21¹ 21¹

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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The function x(u, v) = (R cos(u) cos(v), R sin(u) cos(v), R sin(v)) is a local parametrization of the sphere S of radius R > 0 centered at the origin. Find the components E, F, G of the first fundamental form of S and then use the systems of equations TE+rF = T₁F+G = Eu Fu-Ev T₂E+I₂G T₁₂F+1₂G = = EE+F = F,-/G₁₂ Gu G₁₂F+G Gv Gu to compute the Christoffel Symbols T₁, ₁T2 = ₂1, ²2 = 21, 22, and I'22. - 11: 11' 12 211 12 =
Transcribed Image Text:The function x(u, v) = (R cos(u) cos(v), R sin(u) cos(v), R sin(v)) is a local parametrization of the sphere S of radius R > 0 centered at the origin. Find the components E, F, G of the first fundamental form of S and then use the systems of equations TE+rF = T₁F+G = Eu Fu-Ev T₂E+I₂G T₁₂F+1₂G = = EE+F = F,-/G₁₂ Gu G₁₂F+G Gv Gu to compute the Christoffel Symbols T₁, ₁T2 = ₂1, ²2 = 21, 22, and I'22. - 11: 11' 12 211 12 =
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