Tensor Derivative (continuum Mechanics) - Curl of A Tensor Field - Curl of A Second-order Tensor Field

Curl of A Second-order Tensor Field

For a second-order tensor

$\mathbf{c}\cdot\boldsymbol{S} = c_m~S_{mj}~\mathbf{e}_j$

Hence, using the definition of the curl of a first-order tensor field,

$\boldsymbol{\nabla}\times(\mathbf{c}\cdot\boldsymbol{S}) = e_{ijk}~c_m~S_{mj,i}~\mathbf{e}_k = (e_{ijk}~S_{mj,i}~\mathbf{e}_k\otimes\mathbf{e}_m)\cdot\mathbf{c} = (\boldsymbol{\nabla}\times\boldsymbol{S})\cdot\mathbf{c}$

Therefore, we have

$\boldsymbol{\nabla}\times\boldsymbol{S} = e_{ijk}~S_{mj,i}~\mathbf{e}_k\otimes\mathbf{e}_m$

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