Recent content by mannyfold

Gee, I didn't think that defining e was such a priority. I simply meant that e is the irrational number that serves as the base of the natural log, and as such, it can't be stated as a number with a definite number of decimal places (it is like PI in this regard). Thanks, WigneRacah, you hit...

I'm simply using e = 2.718... and raising it to x, nothing fancy about it. I'm looking for the trick that will give me delta in terms of epsilon (the epsilon-delta proof) or somehow to put a bound on delta. Unfortunately, for this, I'm not at liberty to make definitions. It's not merely...

Can anyone prove that the limit of e^x as x --> 1 is e using the epsilon-delta method? This is not a homework problem, but I am trying to review my course in analysis from a few years back, and this one has me stumped.

Well, "mass" usually means "rest mass" unless otherwise stated. It gets a little hairy in a relativity course, but in real life mass is rest mass.

Hi, I'm not familiar with the theorems, but I can give a few definitions: f is an open mapping if for all open sets O in the domain, f(O) is open in the range. In other words, f maps open sets to open sets. f is a closed mapping if for all closed sets C in the domain, f(C) is closed in...

Keep in mind that Heine-Borel applies only to Euclidean space with the usual topology. It is a special, but important, case in topology.

Thanks guys. I finally got it through my head that df(v) is a function that maps vectors to the reals; hence, dx^i (partial/partial x^j) just maps all the basis vectors to 0 except the ith which it maps to 1 (orthonormality condition). My mistake was that I was still working with the...

One often encounters the following identitiy in Tensor Analysis/Differential Geometry: dx^j (partial/partial x^i) = partial x^j / partial x^i = delta ij It's easy to see why partial x^j / partial x^i = delta ij but how does dx^j (partial/partial x^i) = partial x^j/partial x^i ? I...

Another perspective on this is yet another equality I've found that was not expounded upon: dx^i (partial/partial x^j) = (partial x^i)/(partial x^j) = delta ij I get the last equality. I can't see the first.

Hi Javanse, I appreciate your replies, but I am going to challenge what you wrote. We are seeking to prove df(v) = vf. You wrote: df(v)=(df/dx^i)dx^i(v) What is dx^i(v)? It's simply v^i. But this assumes what we are trying to prove: dx^i(v) = v^jx^i =d/dx^j (x^i) = v^i...

AHA! That is the problem! I don't see how df(v) = vf. Where did you get that definition of the differential? df = partial f / partial x^i dx^i (i = 0 .. n) but the partials here are just the components of the 1-form. I have the feeling that there is something very simple that...

I am having a problem that is glossed over in many textbooks but is driving me nuts. Consider the following inner product or one-form with a vector argument: dx_i(partial_j) = kroenicker delta ij Here, dx_i is a one-form and partial_j (the partial wrt x_j) is a vector. Some books...