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
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 ith component of a vector, and the quantity represents the ipth components of a secondorder tensor.
Therefore, in index notation, the curl of a secondorder tensor can be expressed as
Using the above definition, we get
If , we have
Therefore,
