Let the components of 
be 
, those of
be 
and those of 
be
. Find (i) 
; (ii)
; (iii)
; (iv)
.
:
(2a) show that they are linearly independent.
(2b) Find the components of 
if
(2c) Find the value of 
for
(2d) Determine whether the one- forms 
, 
,
and 
are linearly independent if
,
, 
,
,
, 
,
, 
.
and
be two one forms. Prove by
trying two vectors 
and 
as arguments, that
.
Then find the components of 
.
as the matrix
find (i) the components of the symmetric tensor 
and the antisymmetric tensor 
; (ii) the components
of 
; (iii) the components of
; and (iv) the components of 
.
(4b) For the tensor whose components are 
, does it
make sense to speak of its symmetric and antisymmetric parts? If so,
define them. If not, say why.
(4c) Raise an index of the metric tensor to prove that
tensor, B is a symmetric 
tensor, C is an arbitrary 
tensor, and D is an arbitrary
tensor. Prove:
(5a) 
;
(5b) 
;
(5c) 
.
, the four- vector fields 
and
have the components [ with units where c=1 ]:
and the scalar 
has the value
(6a) Find 
, 
,
. Is 
suitable as a 4-
velocity field? Is 
?
(6b) Find the spatial velocity 
of a particle whose
four- velocity is 
, for arbitrary t. What happens to
it in the limits 
, 
?
(6c) Find 
for all 
.
(6d) Find 
for all 
and 
.
(6e) Show that 
for all 
.
Show that 
for all 
.
(6f) Find 
.
(6g) Find 
for all 
.
(6h) Find 
and
compare with (f) above. Why are the two answers similar?
(6i) Find 
for all 
. Find 
for
all 
. What are the numbers 
the
components of?
(6j) Find 
,
. 
and 
.
If you have any problems please come and see me or contact me by
email.
Peter Dunsby
