Coordinate Transformation (and using line elements)

[darkSun]

This is from Hartle's GR book, in one of the first chapters it talks about diff geom, nothing too advanced, but I am learning on my own.

Homework Statement

It's part E I have trouble with. Read e. and skip to last para if you want.

Consider this coordinate transformation:

x=uv , y=(u^2 - v^2)/2

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

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

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

d. find the equation of a circle of radius r centered at the origin in terms of u and v

e. Calculate the ratio of the circumference to the diameter of a circle using uv coordinates

Homework Equations

The line element in rectangular coordinates, but that's written above

The Attempt at a Solution

Okay, I think curves of constant u and v are vertical parabolas, one facing up and one facing down.

For the line element, I found ds^2= (u^2 + v^2)du^2 + (u^2 + v^2)dv^2

I don't really know how to do c. But that doesn't bother me much,

For the equation of a circle, I got (u^4)/2 + (v^4)/2 = R^2

BUT it's e. I cannot do. I think it is the crux of the question, and I think it's vital that I master this concept. Would it be some sort of line integral using ds over the diameter, and the circumference of the circle? I'm not sure I know how to formulate it correctly. Also not sure what the equation of a straight line in uv coordinates is... Is it v=0?

 

[tiny-tim]

Hi darkSun!

x=uv , y=(u^2 - v^2)/2
…
d. find the equation of a circle of radius r centered at the origin in terms of u and v

e. Calculate the ratio of the circumference to the diameter of a circle using uv coordinates
…
For the equation of a circle, I got (u^4)/2 + (v^4)/2 = R^2

Nooo … you multiplied your 2's wrong … try again!

 

[darkSun]

Oh, I see the error.

So the equation of the circle is (u^4)/4 + (v^4)/4 +((uv)^2)/2 = R^2

But still, how are the line integrals set up? Is it just a matter of setting up the integral in rectangular coordinates then switching to uv coordinates and substituting the Jacobian?

Or is there a way to do it directly from uv coordinates?

Also, for my curiosity, is this what a parabolic coordinate system looks like?

 

[tiny-tim]

So the equation of the circle is (u^4)/4 + (v^4)/4 +((uv)^2)/2 = R^2

= (u² + v²)²/4