steve waterman wrote:Yes. Let's give it a try. Certainly, we need to agree upon what being a Cartesian x co-ordinate means mathematically, as part of any initial set of terms and definitions.
Just to be clear, there's no issue of agreement at this point. We're going to work with the conventional definitions and show how that leads to the Galilean transform. At the end, we can debate whether what we did was the best thing to do or not, but not until then.
Let's get started.
Definition 1: A
manifold is something that everywhere looks locally like Euclidean space. Wikipedia has a
more detailed definition, but this should be good enough for our purposes.
For definiteness, let's consider only 2-dimesional manifolds, things that look locally like the Euclidean plane. The Euclidean plane is obviously a manifold. A sphere and a torus are also two dimensional manifolds, since they look at every point like a point in the plane. And since I just used the word, I ought to introduce
Definition 2: A
point is an element in a manifold. If I want to say that the point P is an element of manifold M, I could write P ∊ M. (But I'll try to avoid doing that and just put things in English.)
If I have two points, P and Q, in a manifold M, the only thing I can say is whether P = Q or not. If they aren't the same point, I can't say anything about the distance or direction between them. The manifold doesn't contain such information. To get it, we need to introduce...
Definition 3: A
coordinate system is a
mapping between ℝ
n and a manifold M. That is, a function that takes n numbers (the
coordinates) and returns a point P in the manifold M. The mapping is
one-to-one, meaning that each set of coordinates maps to a single point in the manifold, and each point in the manifold has only one set of coordinates mapped to it. Not all coordinates need be mapped to the manifold, and not all points in the manifold need to have coordinates.
Since we're dealing with 2-dimensional manifolds, the coordinate system will take a pair of numbers to a point in the manifold. If I call the coordinate system f and the coordinates x and y, I can write f(x,y) = P ∊ M. The mapping being one-to-one means that f(x,y) = f(a,b) if and only if x = a and y = b.
Because the function f is one-to-one, I can invert it to find a function f
-1 that maps M to ℝ
2. That is, f
-1(P) = (x,y) if and only if f(x,y) = P.
Excercise 1: Find f
-1(f(x,y)).
Excercise 2: Find f(f
-1(P)).
This is all dry and boring, so let's have an example to finish this off. Let M be a manifold and f a coordinate system. By marking all the points f(x,0), we can put an x-axis on the manifold. Similarly, we can mark all the points f(0,y) to draw a y-axis. If we do this, we get something like this:
I am purposefully using unusual notation here to emphasize that the coordinates do not live in the manifold -- the
images of the coordinates live in the manifold. This is why I've labeled everything in the form f(x,y).
Example: f(2,1) = P. Therefore, we say that the coordinates of P, in the coordinate system f, are (2,1).
Exercise 3: Draw the point f(-1, 2) = Q on the manifold above.
Exercise 4: Find the coordinates of the point R marked above in the coordinate system f.
I think this is enough for one night. What I need you to do is read this through carefully. If you come to something unclear, stop and ask for clarification -- don't just make up a definition and continue on. Once everything's clear, go through the examples and see that you understand how I got these results. If you don't understand, ask for help. Then do the exercises and post your answers here. Once we're all on the same page, we'll go on to the next step: coordinate transformations. (That should be a bit more interesting that this.)
A note to others: Please don't answer the exercises I've posted for Steve. He needs to work through them for himself to be sure that he really understands the definitions I'm using. You're welcome to offer encouragement, hints, and wry commentary, but please let him do the work.
