Cross product between a vector and a 2nd order tensor

Question

I have been searching for quite a long time, and haven't been able to find any good reference about the cross product between a vector and a tensor: $$ \vec{a} \times \underline{T}= \begin{pmatrix}a_{1} \\ a_{2} \\ a_{3}\end{pmatrix} \times \begin{pmatrix}T_{11} & T_{12} & T_{13} \\T_{21} & T_{22} & T_{23} \\ T_{31} & T_{32} & T_{33} \end{pmatrix}= \quad?$$ I think that I have to take the cross product between $\vec{a}$ and each column of $\underline{T}$ to construct the result, but I would like to be sure. I would also like to know a reference about that. Thank you

Answer 1

Yes, you can certainly define cross products of vectors and 2nd order tensors in Euclidean 3-space.

For example, introduce a cartesian basis, so $\overline{a} \times \overline{T}$ is $$ \overline{a} \times \overline{T} = (a_i \hat{e}^i) \times (T_{pq} \hat{e}^p\otimes \hat {e}^q) \\ =a_i T_{pq} (\hat{e}^i \times \hat{e}^p) \otimes \hat{e}^q $$ where $\otimes$ is the tensor product. If the second step seems unfamiliar, think about what you would do with a dot product $\overline{a} \cdot (\overline{b} \otimes \overline{c})$.

Now we rewrite the cross product itself.

$$ \overline{a} \times \overline{T}\\ =a_i T_{pq} (\epsilon_{lmn} \, e^i_m \, e^p_n \, \hat{e}^l) \otimes \hat {e}^q \\ =a_i T_{pq} \epsilon_{lmn} \, \delta_{im} \, \delta_{pn} \, (\hat{e}^l \otimes \hat {e}^q) \\ =\epsilon_{lip} \, a_i T_{pq} \, (\hat{e}^l \otimes \hat {e}^q) \\ $$ which returns an object with 2 free indices.