Tensor field transformation laws

Let $(𝑀, 𝒜)$ be a $𝐶 𝛼$ -manifold and $𝑋 \in T 𝑝 𝑞 (𝑀)$ be a tensor field. If $𝑥, 𝑥' : 𝑈, 𝑉 \to ℝ 𝑛$ are charts in $𝒜$ , then in local coördinates we have #m/thm/geo/diff

𝑋' 𝜇 1 \dots 𝜇 𝑝 𝜈 1 \dots 𝜈 𝑞 (𝑥') = 𝜕 𝑥' 𝜇 1 𝜕 𝑥 𝛼 1 \dots 𝜕 𝑥' 𝜇 𝑝 𝜕 𝑥 𝛼 𝑝 𝜕 𝑥 𝛽 1 𝜕 𝑥' 𝜈 1 \dots 𝜕 𝑥 𝛽 𝑞 𝜕 𝑥' 𝜈 𝑞 𝑋 𝛼 1 \dots 𝛼 𝑝 𝛽 1 \dots 𝛽 𝑞

Proof

#missing/proof

Often physicists will take this as the definition of a tensor: “A tensor is an object whose components transform as a tensor.”

#state/develop | #lang/en | #SemBr