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. 1. The problem statement, all variables and given/known data 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 2. Relevant equations The line element in rectangular coordinates, but that's written above 3. 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?