Recent content by sal

HallofIvy: No, you missed the point. The question was why the proof which uses hyperreal calculus requires the use of hyperhyper reals (there are two "hypers" there).

For the last time: No, you can't. Regardless of your purpose in looking at this, it's a three dimensional system and you can't just ignore one dimension. The limit is well defined only if it's the same from all angles of approach, and it's not. So, it is not well defined, and in fact if you...

So here it's been four days since I posted the question, 120 people have looked at it, and nobody's attempted an answer. Fooey. In the mean time, though, I think I may have come to a dim understanding of why it must be done this way: In order to even pose the question in a way that makes it...

Just one very quick response to one part of your message: That's not exactly correct. The limit depends on the direction from which you approach the x axis. If you move dy instead, you find the limit is 0, not 1. Ergo it's not well defined, and you cannot conclude that the value for the...

If I found the curl right, then on the x-axis your curl equation is formally 0/0. In short, you've got what looks like a major discontinuity at the x axis. Anyplace off the axis, when z is zero, so's the z component of the curl; ditto for y. Yet on the axis, depending on how you take the...

A uniformly convergent sequence of continuous functions converges to a continuous function. I have no problem with the conventional proof. However, in Henle&Kleinberg's Infinitesimal Calculus, p. 123 (Dover edition), they give a nonstandard proof, and they use the hyperhyperreals to do it. I...

Proofs are fun, and the ones that are the most fun are formalizations of arguments that start out being "visual" -- proof-by-picture. A good proof is often called "elegant", "beautiful", or "lovely", which gives you a good idea of how mathematicians feel about them. The derivation of the...

I talk too much, and this post has gone 'way overboard. I won't be responding in this thread again, unless some very specific question (with a short answer!) comes up. "Analysis", at least where I came from, was multivariate calculus "done over again, right". But I suspect the term may...

I'm a computer programmer who plays with math and physics as a hobby, so take this response with as much salt as you like. I would characterize differential geometry and high school plane/solid geometry as being, on the surface at least, almost completely unrelated fields. If you want a...

After some fiddling, and playing around with alternative matrices for "N", I posted a response based on Gel's comments on fr.sci.physique (with due credit to Gel). The OP appreciated it. Thread may be found here...

Oops -- I posted a full solution here and then realized ... this is a homework problem, isn't it? At least it sure sounds like one. So here's a piece of a solution. Should get you started! It seems like your 1:1 mapping between fixed points in fg and gf generalizes to a 1:1 mapping between...

Thanks -- I think that leads in the right direction. In particular, I was thinking in terms of factors with just two additive terms, and getting nowhere fast trying to come up with generalizations; what you're suggesting leads to something where, for an NxN matrix, each factor would have N...

As the Wiki article says, there's more than one way to extend the cross product to higher dimensions. One very straightforward way extends it to all dimensions of Rn, but it's a little peculiar: In dimension n it becomes a function of n-1 vectors rather than a simple product of two vectors...

Determinant formula in monomials -- can it be generalized? I ran across this question in one of the Usenet groups (fr.sci.maths), and after doing a double take and realizing what was actually being asked I realized I don't know the answer, and after searching a bit I haven't turned it up, so I...

I'm not sure what you mean by "Hamiltonian" in this context, but the Lorentz transforms were surely derived -- or at least written down -- first by Lorentz. That's why they're called "Lorentz" transforms, rather than "Einstein" transforms. Einstein doesn't name the transformation in the main...