Micromechanics of composites/Proof 7

Curl of the transpose of the gradient of a vector

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Let   be a vector field. Show that

 

Proof:

The curl of a second order tensor field   is defined as

 

where   is an arbitrary constant vector. If we write the right hand side in index notation with respect to a Cartesian basis, we have

 

and

 

In the above a quantity   represents the  -th component of a vector, and the quantity   represents the  -th components of a second-order tensor.

Therefore, in index notation, the curl of a second-order tensor   can be expressed as

 

Using the above definition, we get

 

If  , we have

 

Therefore,