Micromechanics of composites/Proof 7
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Curl of the transpose of the gradient of a vector
[edit | edit source]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,