I'm just going to ramble off some observations and thoughts of mine relevant to the question.
First of all, I want to point out that most classes are essentially theoretical (i.e., most classes aren't labs.) These classes are taught by both experimentalists and theorists. So the apparent...
I am considering a second order ODE of the form y''(x) + f(x) y(x) = 0, with boundary conditions that y(x) = 0 at plus/minus infinity. Note that f(x) is complex for my case.
It seems that the standard techniques for numerically solving this problem are (a.) the finite difference method and...
Could you help point me to one of these solutions then? I don't recall ever seeing the time-dependent Schrodinger equation solved for transitions or ionization and I'm not sure where to look...
Nevermind. I just forgot for the red-box FBD you still have to include the weight of the mass, so actually it just says the tension in the top rope is equal to twice mg again. Doh. Too long since I did basic physics.
I have a basic pulley problem which has been troubling me.
Consider the system as shown in the attached picture. We have a massless pulley attached to the ceiling and a mass suspended from one side of the pulley.
If we take the free-body-diagram by 'cutting' where the red box is, we find...
I'm thinking that maybe my suggestion above doesn't work, so let me rephrase/restate my question.
Say we want to define a vector field in R^3. Then we need a way to consistently define a directional basis at each point, and a position for each point. The common choices are well known --...
Say we have a vector field defined in R^3. That is, at every point p in R^3, we have the corresponding set (p, v(p)). In representing this field, as far as I can tell, we have a certain list of very general requirements. That seems to be
a.) an origin,
b.) three everywhere non-coplanar curves...
Wow Fredrik, that's an outstanding response... thumbs up...
Regarding question 3 --
This resource "http://homepages.cae.wisc.edu/~callen/FluxCoordinates.pdf" [Broken] is frequently used in my department for introducing the notions of "covariant components" and "contravariant components"...
Having been through a through a number of textbooks, I can say without reservation Jackson is one of the best books I have studied. Some books are simply incomprehensible (leaving out details in the derivations, omitting crucial facts, speaking in confusing language, speaking unclearly). Some...
OK, thank you for that review. For my own edification I will make a few comments or remarks now. I thought that a chart just had to be smooth -- I didn't realize that we actually only require the weaker condition that the chart only needs to be smoothly compatible.
Also, I suspected that the...
Ah yea, sorry, I meant to say *smooth*, not continuous.
Regarding your answer -- I don't understand, or maybe I am not understanding what it means for this map to "generate" the differentiable structure. As I understand it, this map x^3 will act just like a chart; so therefore it must be a...
I'm not sure what a Peano curve is, but I know a simple case where you can have this to be true. If you take a toroidal curve with an irrational rotational transform (the ratio of poloidal period covered in one full toroidal period), then as the number of transits goes to infinity, this curve...
Consider the manifold of the real-line R with a differentiable structure generated by the map x^3 : M \rightarrow \mathbb{R} . This example is given in a textbook I'm looking at, but I don't really understand how this can work. The inverse map is clearly not smooth at x=0.
Do they mean that...