New Coordinate System

By Brad Rodgers (P1930) on Friday, February 23, 2001 - 09:04 pm :

I'm trying to come up with a new sort of coordinate system. This is going to take a while to explain, so be braced. For example, here is one sort of change of coordinates. Suppose the we made a line, call it b so that on line b, y=x. Then make another line, a so that on line a, y=-x. Now, we can call a and b axis, and put equations in terms of them. Now suppose we were to take this one step further. Suppose that we were to make line b a curve, and line a into an infinite variety of lines, perpendicular to line b. Call the coordinate on axis b the length from any origin (you can choose one for convenience) to a point following the curve. Then call the coordinate on axis a the distance out from curve b. Here's a picture to help:
I've already worked out a few relationships dealing with this, most of which are ridiculously complex, and to determine an a (based upon a b), you have to chose it based upon gradients and then compare two coordinates. Not too nice. Anyways, my question is, has anything like this ever been done before (aside from polar coordinates)? I think this could be used to simplify integrals. Are there any other possible uses. Any simple ways that you can work out that can change coordinates given that b= a given function of x. Sorry if I've made this confusing in my explanation, and I can post my way to change coordinates if need be, but once again, it's quite messy and I don't think it would work all that well in practice.

Brad

By Anonymous on Friday, February 23, 2001 - 09:42 pm :

Yes there is a general theory for change of variable in multi- dimensional integrals which I guess you'd come accross in a first year uni course.

By Brad Rodgers (P1930) on Sunday, February 25, 2001 - 02:31 am :

What is the theory that you describe, anonymous? Do you know where I can find some information on it? What are some of the key formulas in it?

Thanks,

Brad

By Dan Goodman (Dfmg2) on Sunday, February 25, 2001 - 03:04 am :

Brad, the theory I think anon was referring to is the change of variable formula, which says that if

x_{1}

x_{n}

are one set of coordinates, and

y_{1}

y_{n}

are another set, then

\int \int \dots \int F (x_{1}, \dots, x_{n}) {dx}_{1} \dots {dx}_{n} = \int \int \dots \int | det J | F (x_{1} (y_{1}, \dots, y_{n}), \dots, x_{n} (y_{1}, \dots, y_{n}) {dy}_{1} \dots {dy}_{n}

where

x_{i} (y_{1}, \dots, y_{n})

are the change of coordinate formulae from

y

x

, and

J

is the Jacobian matrix of the change of variable formulae, defined as

J_{ik} = \partial x_{i} / (\partial y_{k})

and

det (J)

is the determinant of the matrix

J

As an example, changing between polar coordinates and rectangular coordinates in 2D, you have the formulae:

$x = r \cos (θ)$ , $y = r \sin (θ)$

So $\partial x / \partial r = \cos (θ)$ , $\partial x / \partial θ = - r \sin (θ)$ , $\partial y / \partial r = \sin (θ)$ , $\partial y / \partial θ = r \cos (θ)$ , so the Jacobian matrix

$J = (\begin{matrix} \cos (θ) & \sin (θ) \\ - r \sin (θ) & r \cos (θ) \end{matrix})$

So $det J = r \cos^{2} (θ) + r \sin^{2} (θ) = r$ . The formula then gives us

$\int \int F (x, y) dx dy = \int \int r F (r \cos (θ), r \sin (θ)) dr d θ$

So we could use this to evaluate the area of a circle for example. If we integrate the function $F (x, y) = 1$ over the set $D = {(x, y) : x^{2} + y^{2} \leq 1}$ we get the area $A$ of the circle $D$ . However, this is quite a complicated integral to do using the normal technique, since the limits are quite complicated. If we change variables to polar coordinates however, the limits are easy, $r$ ranges from0 to $R$ and $θ$ ranges from 0 to $2 π$ . So

$\begin{matrix} A & = & \int \int_{D} dx dy = \int_{0}^{2 π} \int_{0}^{R} r dr d θ \\ = & \int_{0}^{2 π} [r^{2} / 2]_{0}^{R} d θ \\ = & \int_{0}^{2 π} R^{2} / 2 d θ \\ = & (R^{2} / 2) [θ]_{0}^{2 π} \\ = & (R^{2} / 2) (2 π) \\ = & π R^{2} \end{matrix}$

Hooray! It works!

By The Editor :

One problem with Brad's coordinate system is that the (a,b) coordinates of a point need not be uniquely determined: there might be several nearest points for instance. And some (a,b) coordinates might never occur.

There are plenty of funny coordinate systems out there, though.