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


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