Continuum mechanics/Curl of a gradient of a vector

Curl of the gradient of a vector - 1Edit

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

 


Curl of the transpose of the gradient of a vectorEdit

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,