Recent content by Doodle Bob

Actually, you apparently didn't get your answer, given the second sentence above. Real analysis, at least at the entry-level, is the adult version of calculus. It is needed to prove all of the results that are typically used in calculus. In fact, that is the reason for its existence: to give...

I'm glad to help but you really should seek out a graduate school somewhere to study this kind of stuff. Human discourse is indispensable in understanding this level of material. In fact, These kind of mysteries are usually resolved after 5 minutes of conversation.

There are two things you need to be careful about when plugging a vector into a 2-form. The first is what you were initially concerned with: whether the author uses the factorial convention or not. In this case, it looks likes the author doesn't. The second, though, is something you've...

Seriously, though, brick-layers also don't generally have to know and understand set theory, but that's not a particularly high mark of honor for the field of masonry. Why you feel the need to denigrate a field just because you don't understand it very well, is beyond me.

Actually, that wikipedia article leaves out some mathematically crucial aspects to the so-called "definition" of pi being the ratio of the circumference of the circle to its diameter. 1. It shouldn't be taken on faith that this is a well-defined concept. After all, such a constant doesn't...

I've never found anything in mathematics to be beautiful. The concept of beauty in mathematics traces back to Hardy's A Mathematician's Apology and is based on a more-or-less Late 19th/Early 20th Century sense of aesthetics. Nevertheless, this equation has always intrigued me, since it gives...

Note: rotations about the origin preserve this metric.

The problem with this argument is that geodesics are not the only curves preserved by isometries. E.g. lines of latitude are also preserved (but not fixed) by certain rotations. A correct proof would need to show directly or indirectly that the great circles are the only curves on the sphere...

...what he said. In fact, you can get a free copy of one of those standard books here: http://www.math.harvard.edu/~shlomo/ (Advanced Calculus) Check out p. 373.

For some reason, I am not convinced that there would countably many segments. But a non-measure theoretic argument that the geodesic will not fill the torus is simply finding a point that won't be on the geodesic: if this is the standard unit square torus and the geodesic is given as a line...

Try Sergei Tabachnikov's 1995 book "Billiards": you can find a preprint here http://www.math.psu.edu/tabachni/Books/books.html

I'm not sure about this. Isn't F(x)= \int_0^x |t|dt differentiable at 0? It is the piecewise function given by F(x)=x^2 for x>0 and F(x)=-x^2 for x>0 and F(0)=0.

I really think you should re-investigate what definition of diffeomorphism you are thinking of here, because as far as I use the term, diffeomorphisms by definition preserve the differentiable structures of the manifolds involved. Certainly, this is true if you are assuming that diffeomorphisms...

Sorry, I misread the lemma. So, now I'm confused as to what the confusion is: diffeomorphic differentiable structures are essentially the same differentiable structure and will share the same differentiable functions -- up to, of course, the amount of differentiability of the diffeomorphism...

BTW the problem with your "proof" is the second sentence above, which is decidedly not true in general. I think you are mixing up the terms "smooth" and "homeomorphic" in much of your narrative.