Relation between axial vector and displacement[edit | edit source]
Let be a displacement field. The displacement gradient tensor
is given by . Let the skew symmetric part of the displacement
gradient tensor (infinitesimal rotation tensor) be
Let be the axial vector associated with the skew symmetric
tensor .
Show that
Proof:
The axial vector of a skew-symmetric tensor satisfies the
condition
for all vectors . In index notation (with respect to a Cartesian
basis), we have
Since , we can write
or,
Therefore, the relation between the components of and
is
Multiplying both sides by , we get
Recall the identity
Therefore,
Using the above identity, we get
Rearranging,
Now, the components of the tensor with respect to a Cartesian
basis are given by
Therefore, we may write
Since the curl of a vector can be written in index notation as
we have
where indicates the -th component of the vector inside the
square brackets.