Del
Tensor derivative

Del can also be applied to a vector field with the result being a tensor. The tensor derivative of a vector field v {\displaystyle \mathbf {v} } (in three dimensions) is a 9-term second-rank tensor – that is, a 3×3 matrix – but can be denoted simply as ∇ ⊗ v {\displaystyle \nabla \otimes \mathbf {v} }, where ⊗ {\displaystyle \otimes } represents the dyadic product. This quantity is equivalent to the transpose of the Jacobian matrix of the vector field with respect to space. The divergence of the vector field can then be expressed as the trace of this matrix.

For a small displacement δ r {\displaystyle \delta \mathbf {r} }, the change in the vector field is given by:
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