Partial derivative as a local affine connexion

Let $(𝑀, 𝒜)$ be a $𝐶 𝛼$ -manifold and $𝑥 : 𝑈 \to ℝ 𝑚$ be a chart in $𝒜$ . Let $(𝜕 𝜇 𝑎) 𝑚 𝜇 = 1$ denote the associated partial derivatives. We can define the ordinary derivative $𝜕$ as an affine connexion local to $𝑈$ #m/thm/geo/diff so that for $𝑋 𝑎 = 𝑋 𝜇 𝜕 𝜇 𝑎 \in 𝔛 (𝑈)$ and $𝑌 𝑎 = 𝑌 𝜇 𝜕 𝜇 𝑎 \in 𝔛 (𝑈)$ we have¹

𝜕 𝑌 𝑋 𝑎 = 𝑌 𝑏 𝜕 𝑏 𝑋 𝑎 = 𝑌 𝜇 (𝜕 𝜇 𝑋 𝜈) 𝜕 𝜈 𝑎 .

Proof

Follows from standard properties of the Partial derivative.

#state/tidy | #lang/en | #SemBr

2009. General relativity, §3.1, p. 32. ↩

Partial derivative as a local affine connexion

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