reach the bottom at
times 
and 
. Prove that, so long as 
(i.e. the rocket moves non- relativistically), one has
is a scalar and
are two vectors.
(2a) Compute f as a function of polar coordinates 
and
find the components of 
and 
on the polar
basis, expressing them as functions of r and 
.
(2b) Find the components of 
in Cartesians and obtain
them in polars by (i) direct calculation in polars, and (ii)
transforming from Cartesian coordinates.
(2c) Use the metric tensor in polar coordinates to find the polar
components of the one- forms 
and 
associated with 
and 
. Obtain these components by
transformation of the Cartesian coordinates of 
and
.
of question (2)
above, compute;
(3a) 
in Cartesian coordinates;
(3b) the transformation 
to polars;
(3c) the components 
using the
Christoffel symbols given in the lecture notes [ Why is this the same as
(3b)?];
(3d) the divergence 
in Cartesian coordinates;
(3e) the divergence 
in polars
using part (3c);
(3f) and the divergence 
using
the formula given in the lecture notes.
for all possible
indices in polar coordinates for the tensor with polar components
(
,
, 
,
)
If you have any problems please come and see me or contact me by
email.
Peter Dunsby
