Knowlpedia

Definition

Row divergence of a matrix field

The vector obtained by differentiating a matrix field in its second index.

For a differentiable TT, its row divergence is the vector

(div⁡T)i=∑j∂xjTij.(\operatorname{div}T)_i=\sum_j\partial_{x_j}T_{ij}.

Thus each row is treated as a vector field. Some conventions differentiate the first index; the two agree for symmetric tensors but must be distinguished otherwise.

Outer-product calculation

With (v⊗w)ij=viwj(v\otimes w)_{ij}=v_iw_j, the product rule gives

div⁡(v⊗w)=(w⋅∇)v+vdiv⁡w,((w⋅∇)v)i=∑jwj∂jvi.\operatorname{div}(v\otimes w)=(w\cdot\nabla)v+v\operatorname{div}w, \qquad ((w\cdot\nabla)v)_i=\sum_jw_j\partial_jv_i.

In particular, a divergence-free velocity satisfies div⁡(u⊗u)=(u⋅∇)u\operatorname{div}(u\otimes u)=(u\cdot\nabla)u. For T=pIT=pI, row divergence is ∇p\nabla p.