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 -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,