Tensor field

Tensor field transformation laws

Let (𝑀,𝒜) be a 𝐶𝛼-manifold and 𝑋 ∈T𝑝𝑞(𝑀) be a tensor field. If 𝑥,𝑥′ :𝑈,𝑉 →ℝ𝑛 are charts in 𝒜, then in local coördinates we have diff

𝑋′𝜇1⋯𝜇𝑝𝜈1⋯𝜈𝑞(𝑥′)=𝜕𝑥′𝜇1𝜕𝑥𝛼1⋯𝜕𝑥′𝜇𝑝𝜕𝑥𝛼𝑝𝜕𝑥𝛽1𝜕𝑥′𝜈1⋯𝜕𝑥𝛽𝑞𝜕𝑥′𝜈𝑞𝑋𝛼1⋯𝛼𝑝𝛽1⋯𝛽𝑞

Proof

proof

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

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