Recent content by jackferry

I realized that I thought the chart had to be differentiable, but it is the transition map between charts that must be differentiable, which makes more sense

I was reading about differentiable manifolds on wikipedia, and in the definition it never specifies that the differentiable manifold has a metric on it. I understand that you can set up limits of functions in topological spaces without a metric being defined, but my understanding of derivatives...

A question I've had about the black body problem and classical physics in general has to do with the conservation of energy. One of the first things you can derive in classical mechanics is that for a conservative force the total energy of the system doesn't change. However, one of the typical...

So I made this diagram of the order imposed on the set of topologies on a 3 point set. Each small circle is a topology with the circle elements being the open sets. The larger circles join isomorphic topologies together. The ordering was messy when I tried to draw inclusion from each topology to...

I was listening to this lecture: and in it, sometime around the 30:00 to 40:00 minute mark, he implies that the torus' sturcture built up from the orbits of the group under addition on the real plane is the same idea as the cylinder's structure being built up from the orbits of the group under...

That's amazing! So a slightly tangential question: when we talk about problems, esp. Differential equations, I've heard something along the lines of "there is no general analytical solution". Why is that important? Are non-analytical solutions difficult to find? Is the challenge somehow due to...

Would it be possible to extend that cardinality to smooth functions somehow, or are those too complex?

Awesome, thank you very much! So this means that we can enumerate all real analytical functions by choosing a point, and "listing" all the possible values of the function at that point, and then doing the same with all of its derivatives. This seems to imply that the order of analytical...

Does this imply that they will be equal for their entire domain?

Hello, I am learning about smooth analytic functions and smooth nonanalytic functions, and I am wondering the following: Is there a theorem that states that for any real analytic functions f and g and a point a, that if at a f=g and all of their derivatives are equal, that then f=g?

I'm an undergraduate taking a physical chemistry course, and I got to a part in my reading about the derivation of the ideal gas law. The passage is linked below...

Are the D's necessary in a cyclotron? It seems to me that the same effect could be achieved without the discontinuity, which would eliminate the need for a high operating voltage. If the D's are replaced by a complete conducting circle of metal, you could just apply the alternating voltage to...

I'm an undergrad physics student trying to wrap my head around basic QM ideas, and the question I had was this: when we talk about the energy levels of an atom and the wavefunction of the electron around that atom, we talk about an electric potential that affects the shape the wavefunction...

Thank you very much, this makes much more sense. I was getting very confused and a little annoyed about the whole 1+2+3+4+... = -1/12 thing, but it helps a lot to have some context.

I was reading the Wikipedia article about the sum 1+2+3+4+..., and I saw this explanation: c = 1+2+3+4+5+6+... 4c = _4__+8__+12+... -3c = 1-2+3-4+5-6+... link: http://en.wikipedia.org/wiki/1_+_2_+_3_+_4_+_%E2%8B%AF My question, as one who hasn't worked with infinite sums: Why are you...