Coordinates in geometry

  • #1
Hello there.Could coordinates be functions?For example in a n-manifold with (x1,...xn) let be the coordinates could they be functions of a coordinate system not belonging to the n-manifold?Or we could first use a coordinate system then have our results, and then have a second coordinate system and use the coordinates of the first system as variables in each coordinate of the second to use them as functions?For example let us have the cartesian coordinate system (x1,...xn) and the second as (f1(x1,...xn),...,fn(x1,...xn)), could this happen?Also, could we use on two different pieces of a manifold different coordinates?Thank you.
 

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  • #2
Stephen Tashi
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Hello there.Could coordinates be functions?
What definition are you using for term "coordinates"?

Also, could we use on two different pieces of a manifold different coordinates?
Have you studied the definition of a "manifold"? - atlas? - charts?
 
  • #3
PeroK
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Hello there.Could coordinates be functions?For example in a n-manifold with (x1,...xn) let be the coordinates could they be functions of a coordinate system not belonging to the n-manifold?Or we could first use a coordinate system then have our results, and then have a second coordinate system and use the coordinates of the first system as variables in each coordinate of the second to use them as functions?For example let us have the cartesian coordinate system (x1,...xn) and the second as (f1(x1,...xn),...,fn(x1,...xn)), could this happen?Also, could we use on two different pieces of a manifold different coordinates?Thank you.
You mean like polar coordinates: ##x = r\cos \theta, \ y = r \sin \theta##?
 
  • #4
What definition are you using for term "coordinates"?


Have you studied the definition of a "manifold"? - atlas? - charts?
Yes, as I remember I have but I do not know what they mean.
 
  • #5
mathwonk
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A manifold is a geometric object M made up of "points". A coordinate is a number (or sequence of numbers) assigned to such a point. Thus a coordinate is a function defined on some (open) subset of the points of M, with values which are numbers. Thus given a point in the domain subset of a given coordinate, one can ask what is the coordinate of a point in that subset, i.e. what numbers are assigned to that point by that coordinate function.

One can also ask the inverse question, namely given a number (or sequence of numbers), one can ask what point in the domain subset of a given coordinate, is the point having the given coordinate. This views the inverse function of a coordinate function also as a function, namely a function from sequences of numbers, back to geometric points. In particular, both coordinates and their inverses are viewed as functions, one goes from points to numbers, and the other goes from numbers to points.

Because of the complicated shape of the manifold, the domain subset of a given coordinate is usually not the whole manifold, so it is usually necessary to use more than one coordinate function, in order to cover the whole manifold. In this case it often happens that the domains of two different coordinate functions f,g can overlap, so that the same point p of M can be assigned two different coordinates, one by one coordinate function f and another by a second coordinate function g. In this case one can ask another question, given the coordinate f(p) = x, of the point p assigned by the coordinate function f, what is the coordinate g(p) = y, of that same point but assigned by the coordinate function g? This yields a composition function y(x), from numbers to numbers. I.e. if we have two coordinate functions defined near the same point, then we do get a composition which expresses one coordinate as a function of the other coordinate. We can also do this in the other order, namely if we have only one coordinate function f, defined near p, with value f(p)= x, and if we also have any function from numbers to numbers, defined near x, say y(x), we can compose these and obtain a new coordinate function g(p) = y(x) = y(f(p).

Thus we have several types of functions coming from coordinate systems on a manifold, namely functions from points to numbers, called "coordinates", and then we have their inverses, functions from numbers to points, called "parametrizations", and finally we have functions from numbers to numbers obtained by composing parametrizations with coordinates, called "coordinate changes".

E.g. in the upper right quadrant of the plane, we can use either cartesian coordinates (x,y) which are functions from point to pairs of numbers, or we can use polar coordinates (r, theta), which are also functions from points to pairs of numbers, or we can compose these and get coordinate change functions, such as x = r.cos(theta), y = r.sin(theta), or theta = arctan(y/x), r = sqrt(x^2+y^2).

In this example, we do not need more than one coordinate system since the domain of our coordinate functions is in both cases the whole quadrant, and a more typical example can be made using the unit circle x^2+y^2 = 1. I.e.although the pair of numbers (x,y) does suffice to describe any point of the circle, we prefer to have only one number as a coordinate on the circle, since the circle is only "one dimensional". But if we use only one of the two coordinates, say either x or y, we get confusion at certain points (namely near the points (1,0) and (-1,0) for x, and near (0,-1) and (0,1) for y). So we could use 4 coordinate functions, say x on the arc of the circle above the x axis, again x on the arc below the x axis, and we could use y on the arc either to the right or to the left of the y axis.

Notice here the "same" coordinate, namely x, is used for two different subsets of the circle. It really is not the same in the two cases, because part of the data of a function is its domain, and these two uses of x have different domains. Thus when we ask the question, which point of the circle has a certain x coordinate? we have to specify on which semicircle the point lies. So we can ask which point of the upper semicircle has x coordinate sqrt(2/3), and the answer is p = (sqrt(2/3), +sqrt(1/3)). So the inverse of this x coordinate is the parametrization taking x to p = (x, +sqrt(1-x^2)), and maps the x interval (-1,1) onto the upper semicircle. The inverse parametrization of the other x coordinate takes x to (x, -sqrt(1-x^2)), and takes the same x interval (-1,1) onto the lower semicircle.

The coordinate change question might ask: what is the y coordinate of the point in the upper right quadrant of the circle having x coordinate equal to x, and the answer would be y = sqrt (1-x^2). The y coordinate of the point in the lower right quadrant semicircle would be then y = -sqrt(1-x^2).
I hope I got these more or less correct, but you see it gets confusing.

Oh yes, a coordinate function is also called a "chart", and a collection of them with domains covering the manifold is called an "atlas". A (topological) manifold is a topological space equipped with an atlas, but not every topological space has one. I.e. each chart is required to be continuous with continuous inverse, so the topological space has to have an open covering by sets that look topologically the same as open sets in euclidean space. Such spaces are sometimes called "locally euclidean". Some authors put additional technical conditions on a manifold, such as "Hausdorff", "2nd countable", "paracompact", in order to be able to carry out certain global constructions, such as to construct "partitions of unity".

Oh yes, since this is in the differential geometry forum, I assume you want also to do calculus on your manifolds, so you must assume that in your atlas, all coordinate change maps are differentiable or even smooth, so that the differentiability conclusions you reach using two different coordinate functions give the same "qualitative" answers. I.e. you don't get the same numerical derivative with both coordinates, but the question "is this function differentiable?" has the same answer in both.

The following website, although apparently not "secure", has a discussion similar in level to this one:

http://bjlkeng.github.io/posts/manifolds/
 
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  • #6
To change coordinates we can put on each coordinate whatever combination of fuctions like sinx, logx, e^x, function series etc, that we want and we get valid results?
 
  • #7
mathwonk
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not quite. coordinates are supposed to determine the points of their domain uniquely, so if (x,y) are coordinates in a domain D in the manifold, with values running over a coordinate set U in the (x,y) plane, and we want to change to coordinates in the (z,w) plane, the functions z(x,y) and w(x,y) that we use must be invertible functions of x and y. I.e. the function taking (x,y) to (z,w) must be invertible.

In technical language, the coordinate function (x,y) defines a homeomorphism from D in the manifold, to U in the (x,y) plane, and then our functions (z,w) must also define a homeomorphism from U to some other subset V in the (z,w) plane.

I.e. "coordinates" in a subset D of our manifold M, must give a homeomorphism from D to some open subset U of euclidean space. As long as the composition (z(x,y), w(x,y)) does this, then the composition does give a new coordinate system.

E.g. the coordinate x on the lower semicircle of the unit circle, does give a coordinate chart on that semicircle, with coordinate values in the interval (-1,1). But if we compose x with cosine, then the function cosin(x) is not a coordinate on that semicircle, because cosine is not a homeomorphism from the x interval (-1,1) to any other interval. the problem is that since cosine is an even function, cosine is not injective near x=0. The function Sine would do fine however on this interval, since sin(x) is a homeomorphism from the x interval (-1,1) to the interval (sin(-1),sin(1)).

Of course if you want a differentiable manifold, the coordinate change functions should be also differentiable homeomorphisms with differentiable inverse.

You really ought to spend some time now reading the definitions of these terms somewhere and thinking about them.
 
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