I was studying the angular momentum equation in the continuum case and I encountered this identity. I am not sure how the identity is derived. Could some one supply more details and intermediate step? $$\int_s -\vec{n}\cdot(\underline{\sigma}\times\vec{r})dS=\int_s -\vec{n}\cdot\underline{\sigma}\times\vec{r}dS = \int_s \vec{r}\times\vec{n}\cdot\underline{\sigma}dS$$ where $\vec{n}$ and $\vec{r}$ are first-order tensors (vectors) and $\underline{\sigma}$ is a second-order tensor.
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$\begingroup$ What is the cross product of a tensor and a vector? $\endgroup$– Vincenzo TibulloDec 26, 2022 at 10:04
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$\begingroup$ @VincenzoTibullo You can have a look here math.stackexchange.com/questions/1307835/… $\endgroup$– crostataJan 2 at 10:45
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I finally found an answer. Basically, there is no order in the operation, thus one can see the dot product as a single object. The anticommutative property odds. Calling $\vec{n}\cdot\underline{\sigma}=\vec{a}$ then: $$ -\vec{a}\times\vec{r}=\vec{r}\times\vec{a}$$