- For the two vectors
and
,
find
, the component of along , the angle between and ,
, and
.
- The gradient operator
transforms like a vector.
Use this to write the equations of electromagnetism with the Einstein summation convention, that is, with no vector symbols.
To make it interesting, here are the EM equations in Rationalized Heaviside-Lorentz units.
- Use the operator
to show that the divergence of a curl is zero.
- Use the totally antisummetric tensor to derive the identity
.
- Calculate the rotation matrix for a passive rotation about the axis through an angle followed by
a rotation about the new axis by an angle .
- Calculate the rotation matrix for a passive rotation through an angle of 120 degrees about
an axis making equal angles with the original three coordinte axes.
- Calculate the time derivative of a rotation matrix through an angle about the axis.
Remembering that the axial vector
is related to an antisymmetric tensor,
relate the time derivative of the rotation matrix (evaluated at ) to
.
- Show that for matrices
and
,
,
and that
.

Jim Branson 2012-10-21