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 $𝑈$ 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.

tidy | en | SemBr

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

Partial derivative as a local affine connexion

Graph View

Backlinks

Partial derivative as a local affine connexion

Footnotes

Graph View

Backlinks