A tensor derivation is a real-linear operation preserving every tensor type, obeying the tensor-product Leibniz rule, and commuting with every tensor contraction. A derivation of smooth functions and a real-linear operator on vector fields satisfying extend uniquely to a tensor derivation. On a differential one-form it must satisfyThis expression is linear over smooth functions in . In a local frame , the dual rule is . Apply the product rule to every coefficient and frame factor to define the extension; these dual signs cancel under contraction. Frame changes agree by differentiating the inverse matrix. Cutoffs prove locality and hence uniqueness from local expansions.
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