Coordinate Transformation (and using line elements)

In summary, the homework statement is that Hartle's GR book talks about coordinate transformation, specifically the attempt at a solution for finding the equation of a circle in terms of u and v coordinates. The attempted solution is finding the equation of a vertical parabola in terms of u and v coordinates, and that this is what a parabolic coordinate system looks like.
  • #1
darkSun
53
0
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?
 
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  • #2
Hi darkSun! :smile:
darkSun said:
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! :wink:
 
  • #3
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?
 
  • #4
darkSun said:
So the equation of the circle is (u^4)/4 + (v^4)/4 +((uv)^2)/2 = R^2

= (u2 + v2)2/4 :wink:
 

Related to Coordinate Transformation (and using line elements)

1. What is coordinate transformation?

Coordinate transformation is a mathematical process used to convert coordinates from one coordinate system to another. This is often necessary when working with different spatial reference systems, such as converting from latitude and longitude to UTM coordinates.

2. Why is coordinate transformation important in science?

Coordinate transformation is important in science because it allows us to accurately compare and analyze data collected from different sources or in different coordinate systems. It also allows us to visualize and understand spatial relationships and patterns in data.

3. How is coordinate transformation performed?

Coordinate transformation is performed using mathematical equations and algorithms that take into account the differences between the two coordinate systems. This may involve converting between different units of measurement, adjusting for different reference points or datums, and applying rotation, translation, and scaling transformations.

4. What are line elements in coordinate transformation?

Line elements, also known as geodetic lines, are imaginary lines used to represent the shortest distance between two points on a curved surface, such as the Earth. In coordinate transformation, line elements are used to approximate the shape and size of the Earth's surface, allowing for more accurate calculations and transformations.

5. How do errors in coordinate transformation occur?

Errors in coordinate transformation can occur due to various factors, such as inaccuracies in the coordinate systems being used, incorrect parameters or assumptions in the transformation equations, and limitations of the mathematical models used. It is important to carefully consider and account for these potential errors when performing coordinate transformations.

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