Spherical coördinates in differential geometry

Convention

This Zettel uses the Physics convention

The standard global coördinate chart for $ℝ 3$ is given by the identity map. We call these coördinates $(𝑥, 𝑦, 𝑧)$ . Spherical coördinates are better suited to situations with spherical symmetry. Let $𝑈 = ℝ 3 ∖ {0}$ . The coördinate chart $(𝑟 𝐼) = (𝑟, 𝜗, 𝜑) : 𝑈 \to ℝ 3$ is defined so that

𝑥 = 𝑟 s i n 𝜗 c o s 𝜑 𝑦 = 𝑟 s i n 𝜗 s i n 𝜑 𝑧 = 𝑟 c o s 𝜗

The metric is

đ 𝑠 2 = d 𝑟 2 + 𝑟 2 d 𝜗 2 + 𝑟 2 s i n 2 𝜗 d 𝜑 2

so we see the coördinate basis is orthogonal but not orthonormal. Thus we can adjust to get a vielbein

ˆ 𝑟 1 𝑎 = ˆ 𝑟 𝑎 = 𝜕 0 𝑎 ˆ 𝑟 2 𝑎 = ˆ 𝜗 𝑎 = 1 𝑟 𝜕 1 𝑎 ˆ 𝑟 3 𝑎 = ˆ 𝜑 𝑎 = 1 𝑟 s i n 𝜗 𝜕 3 𝑎

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Spherical coördinates in differential geometry

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