Micromechanics of composites/Proof 6

Curl of the gradient of a vector - 1

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Let   be a vector field. Show that

 

Proof:

For a second order tensor field  , we can define the curl as

 

where   is an arbitrary constant vector. Substituting   into the definition, we have

 

Since   is constant, we may write

 

where   is a scalar. Hence,

 

Since the curl of the gradient of a scalar field is zero (recall potential theory), we have

 

Hence,

 

The arbitrary nature of   gives us