Recent content by mathwonk

It seems indeed this is the usual meaning of trivial, i.e. equivalent to a product, but without a choice of trivialization. I was assuming otherwise, but it was just what seemed plausible to me, not citing any source. I should have remembered my own frequent advice not to disagree about the...

well yes. I had assumed you meant this to be an F-bundle over A, in which case those two projections give, in my opinion, one trivialization of it as such. but yes, they also allow it to be considered as a trivial A- bundle over F, which can be thought of as a second trivialization. You just...

I essentially agree with you @cianfa72. i.e. technically a trivial bundle is a trivializable bundle plus a trivialization. the difference however does not matter for many purposes. more generally, a complex line bundle is a locally trivializable C- bundle. most of the time the properties we...

yes thank you. I probably should have said k = real numbers in my discussion above.

Given any (finite dimensional) vector space V, it has a dual space V* consisting of linear functions f:V-->k, where k is the field of scalars. (Take k = real numbers, so we can take square roots to define length.) The most natural feature of V associated to such an f, is the "kernel" of f, i.e...

Yes. If the question referred to the center of mass of a hemispherical shell, (embedded in 3 -space), then since as I believe Archimedes knew, the surface area of a hemisphere equals that of the circumscribing cylinder, and the same holds for horizontal bands cut from the two figures by pairs...

I hope from the comments here it is ok to post a solution. this can be done by noting that, (just as one gets a solid 3 ball by revolving a half disc around its straight edge), revolving a (solid) hemisphere (i.e. a half 3-ball) around the plane of its equator in 4 space, gives a solid 4 ball...

Here is what I learned from reading Riemann. Recall the Riemann- Roch theorem is a formula for the vector dimension of the space L(D) of meromorphic functions on a compact connected complex manifold M of complex dimension one (i.e. a "Riemann surface"), having poles only at most at a given set D...

@martinbn: Riemann's original discussion is reproduced on pages 105-108 of: the dover reprint of Riemann’s works in the original German: especially sections 4,5 of part I of his paper on abelian functions. In the following link, it occurs on pages 19-22, or more fully on 15-22 of the pdf file...

I think vela hit the nail on the head. Becoming educated, i.e. learning to understand something, is for me a long, difficult process requiring significant effort from the learner. A friend who teaches reading emphasizes to her students the need to “engage their thinking mind”. One receives...

As a high school student long ago in the US south our school offered no calculus, very inadequate algebra and geometry, and the local college ridiculed a request to enroll in a class there as a high school senior. There were no supplementary programs available, to my knowledge. So I amused...

Here is the situation as I understand it. Stillwell states Euclid's parallel postulate, then gives an argument meant to prove its converse, and then states that "thus" the parallel postulate implies its own converse. One would expect this to mean that his argument has used the parallel...

yes, my drawing labeled the angles oppositely to Stillwell. I have (hopefully) corrected the notation in response to your observation. As to rigor, such arguments require a rigorous discussion of sides of a line, and what happens when a line meets and hence crosses to the other side of a line...

But let's try to prove existence of parallels actually using Euclid's parallel postulate. let p be the point where n meets m in his diagram 1.2, and let q be the point where n meets l, and assume alpha +beta = π. If l meets m on the right side, say at x, then choose a point, say y, further out...

Having found and read Stillwell's account in full, I agree with you that he seems mistaken in his claim. I find that very puzzling, as he has a PhD in logic from MIT under Alonzo Church, and I wonder if I have understood him correctly.