For coordinates (left(x^{1}, x^{2} ight)) and metric (g=operatorname{diag}left(g_{11}, g_{22} ight)), the Gaussian curvature is For a sphere


For coordinates \(\left(x^{1}, x^{2}\right)\) and metric \(g=\operatorname{diag}\left(g_{11}, g_{22}\right)\), the Gaussian curvature is

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For a sphere with coordinates defined in the following figure,

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show that a Gaussian curvature \(K=R^{-2}\) is obtained using spherical coordinates \(\left(x^{1}, x^{2}\right)=(S, \phi)\) or cylindrical coordinates \(\left(x^{1}, x^{2}\right)=(r, \phi) .

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