| 21. | On a manifold, a tensor field will typically have multiple indices, of two sorts.
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| 22. | The covariant derivative of a tensor field is presented as an extension of the same concept.
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| 23. | Nowadays, one recognizes this as a single antisymmetric 2nd-rank tensor field in spacetime.
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| 24. | The divergence of a continuously differentiable second-order tensor field is a first-order tensor field:
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| 25. | The divergence of a continuously differentiable second-order tensor field is a first-order tensor field:
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| 26. | Suggestively, replacing the vector field with a rank-tensor field, this can be generalized to:
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| 27. | In general, one can define various divergence operations on higher-rank tensor fields, as follows.
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| 28. | More precisely, a tensor field assigns to any given point of the manifold a tensor in the space
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| 29. | This handles the formulation of variation of a tensor field " along " a vector field.
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| 30. | The components of this derivative of a tensor field transform covariantly, and hence form another tensor field.
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