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