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Topic: Analytic vs. Synthetic Geometry
Replies: 7   Last Post: Sep 5, 2013 10:55 PM

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Posts: 134
Registered: 2/21/06
Re: Analytic vs. Synthetic Geometry
Posted: Sep 5, 2013 10:55 PM
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On Thursday, September 5, 2013 10:03:28 AM UTC-4, Peter Percival wrote:
> Lite Beta wrote:

> >
> > Also, R^n is the Cartesian product. So isn't it by default that the n numbers in R^n are cartesian coordinates?
> It seems to me that "R^n" is ambiguous, it might be nothing more than
> this set: {(x_1,...,x_n): x_i real, 1 <= i <= n}. Or it might be that
> set with structure on it. In the latter case, there are various
> possible structures: additive group, vector space, metric space,...

So do you think cartesian coordinates are not more closely linked to R^n than other coordinate systems? The wikipedia page on Euclidean spaces mixes the two and reading it scrambled (or rescrambled) my brain.

How about this:
If you have a constant function f from a subset S of R^2 into R, and you do the double integral of f*dA over S, that is interpreted as the volume of a box with base area S and height f. But doesn't this require us to consider R^2 as the cartesian coordinates (x,y) ?

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