"Zinc Potterman" . (delete 123's to reply) wrote
in message ...
I am still trying to self teach Gen Rel with Hartle's Gravity...
Can someone please check my maths.

(x,y) coordinates have to be transformed to new (u,v) coordinates using

x=uv and y=½(u^2-v^2)

a. Sketch curves of constant u and constant v in xy plane

I got parabolae of form y= +/-ax^2

b.Transform the line element dS^2 = dx^2 + dy^2 into (u,v) coordinates

I got dS^2 = ¼(du^2 + dv^2)^2

c. Do the curves of constant u and constant v intersect at right angles?

My parabolae mentioned earlier are positive for uv and negative for u,v
and
only meet at
the origin.

d.Find the equation of a circle of radius r in terms of u and v

I got
r^2 = ¼(u^2 + v^2)^2
r = ½(u^2 + v^2)

e.Calculate the ratio of the circumference to the diameter of a circle
using
(u,v)
Do you get the correct answer?

I tried a line integral of dS from part b, but got stuck here as it
probably
needs a substitution.

I got (so far) Circ = Integral dS = Integral ½(du^2 + dv^2)

Love to get hold of the solutions manual so i didn't have to keep asking.
Thanks
Zinc


I sent you a pdf showing the solution but never received a reply.

Your result for dS^2 is incorrect.

It is possible to intergrate half way around the circle using uv
coordinates. The result is that the circumference is 2 Pi r. This is what we
would expect because the xy plane is flat and this is just the introduction
of new coordinates.

It is also possible to calculate the Riemann tensor for the uv metric and it
is identically zero, just confirming that the xy plane is flat!

David Park

http://home.earthlink.net/~djmp/