# Search results

1. ### Learning Physics from listening to ipod?

Listening to the Feynman lectures or something similar is a very good idea. But please let this only be a secondary resource and not your main resource for learning. If you don't only rely on this, but are focused on the math and problem solving, then this could be a very good thing.
2. ### Spring with 2 masses free fall

Thank you very much haruspex! I was doing the entire problem with the mindset of someone analyzing an inverse-square gravitational field such as that exterior to a sphere of uniform density, although I have absolutely no idea why I was stuck in this mindset :redface: you'll have to excuse the...
3. ### Spring with 2 masses free fall

Oh sorry I meant to say the opposite of what I actually said. I was thinking of the field external to the sphere of uniform density. Inside it varies linearly with the distance from the center of the sphere of uniform density.
4. ### Spring with 2 masses free fall

Thank you so much mfb. If we let ##u = r_a - r_b - l## in the equation ##m(\ddot{r}_a - \ddot{r}_b) = -2k(r_a - r_b - l) + f(r_a - r_b)## then we get ##m\ddot{u}+ (2k - f)u = fl##. Wouldn't this imply that the new period is ##T = 2\pi \sqrt{\frac{m}{2k - f}}## as opposed to ##T = 2\pi...
5. ### Looking for text on stochastic processes

What about Feller's excellent two volume set on probability? https://www.amazon.com/dp/0471257087/?tag=pfamazon01-20&tag=pfamazon01-20 Especially the second volume seems to have some good stuff. There might also be an introduction to measure theory in the book.
6. ### Spring with 2 masses free fall

I'm not sure I follow, at least with regards to the physical system in this problem. We have a spring with two identical masses one on each end and if the system is allowed to oscillate freely then the period should be ##T = 2\pi \sqrt{\frac{m}{2k}}##. See for example ehild's post here...
7. ### Spring with 2 masses free fall

Thanks for the reply! As for the tidal forces having the same sign: if we imagine the center of mass as a hypothetical particle relative to whom both the top and bottom mass experience tidal forces then shouldn't both the top and bottom mass be accelerating away from the center of mass under...
8. ### Spring with 2 masses free fall

Homework Statement The Attempt at a Solution The reference to exercise 1.7 is not essential. The only part exercise 1.7 that is relevant to this exercise is the following: if a long narrow tube is drilled between antipodal points on a sphere of uniform mass density and two identical...
9. ### Convulation and integration

But ##\int f(x) dx## and ##\int g(x)dx## are just real numbers, no? What is the convolution of two numbers?
10. ### Compactness and Connectedness

See http://planetmath.org/proofofheineboreltheorem under the heading "The case n=1, the closed interval is compact". This contains an elementary proof of the fact that closed intervals in ##\mathbb{R}## are compact. If you already know the Heine-Borel theorem however, then you know that...
11. ### Disjoint Cycles

If you have one cycle, then you can find the inverse by reversing the cycle. So if \sigma = (1 ~2~5~3), then \sigma^{-1} = (3~5~2~1) Then if you have a more general form, then you can calculate the inverse by the formula ##(\sigma\tau)^{-1}= \tau^{-1}\sigma^{-1}##. For example, if you have...
12. ### Compactness and Connectedness

What is your definition of compact? Right!
13. ### Compactness and Connectedness

Yes to both. Try to find counterexamples.
14. ### Disjoint Cycles

How did you define ##\pi^\sigma##? Did you define it as ##\sigma \circ \pi \circ \sigma^{-1}##?
15. ### Two exercises on complex sequences (one about Mandelbrot set)

You have |z_0| = 0 |z_1| = |c| |z_2| = |z_1^2 + c| \ |c| |c+1|\geq |c| |z_3| = |(c^2 + c)^2 + c| = |c^2 (c +1)^2 + c| = |c| |c(c+1)^2 + 1|\geq |c|( |c|^2 |c+1|^2 - 1) \geq |c| (|c|^2 -1) \geq |c|^2 where I make fundamental use of the inequality ##|a-b| \geq ||a| - |b||##.
16. ### Function that is an isomorphism

The theorem says that there is some function that is an isomorphism. It does not say that any function ##T## is an isomorphism. In your example, the theorem is satisfied since ##S(x) = x## is an isomorphism. But the theorem does not say that any arbitrary ##T## is an isomorphism. So you can...
17. ### Contour integration confusion

Let ##\alpha_r## be the arc with radius ##r##. You need to prove \lim_{r\rightarrow +\infty} \int_{\alpha_r} \frac{1}{(x^2 + 1)^2}dx = 0 Or you could prove the same thing with absolute values. Note that \frac{|(x^2 + 1)^2|}{|x^3|}\rightarrow +\infty So there exists a ##C## such that...
18. ### Can you assess my course load this semester?

It's hard to answer this for you. Some will be able to handle it, others don't. Have you tried semesters with 5 classes already? How did you do? Did you make some kind of study schedule (saying how many hours each day/week you would spend on what)? That said, topology and analysis complement...
19. ### The set of the real numbers is closed

Yes, it is closed and it is open. Therefore I found it a bit weird that you said that the empty set is not open.
20. ### Two exercises on complex sequences (one about Mandelbrot set)

Yes. Note however that ##|r^n e^{ni\theta}| = r^n##. This is a better approximation. But your method was good too. Yes. Take ##|c|>2##. Maybe you could start with writing out the first 5 or 6 terms of the ##z_n## sequence to see if you can find some pattern. Of course you only care about...
21. ### The set of the real numbers is closed

How is the empty set not open?
22. ### The set of the real numbers is closed

Well, if we're going to nitpick: the statement should read "##\mathbb{R}## with the Euclidean metric is a completion of the rationals" Since there are multiple topologies on ##\mathbb{R}##
23. ### Vector Problem

But ##\sqrt{a^2} = |a|##. So if you pull out this, then you get \cos(\theta) = \frac{a}{|a|}\frac{1 + \frac{6}{a}}{\sqrt{1 + \frac{4}{a^2}}\sqrt{10}} So you need to decide what ##\frac{a}{|a|}##. This will be ##1## or ##-1## depending on whether ##a## is positive or negative. But since you...
24. ### Criteria for instructive problems in self-study

In my opinion, the proof-type questions are the most valuable type of question. They really test your understanding of the concepts. The computation type questions are of course still important. You should know how to find eigenvalues and diagonalizations. But once you solved a couple of them...
25. ### Math Class To Self-Study?

If you already know calculus, then you have a ton of options. Since you already done calc III and you're a junion, I'll assume you want books that are pretty challenging. Here are some things you could do: Linear algebra (but you already will do that, judging from your posts) Abstract...

Good luck!!
27. ### Any good books on PDEs?

I doubt it, but you should try anyway. If it works out then you found a very good book, otherwise you know what you need to work at. You sure this is legal? You might want to remove it before the mentors see it :tongue: But I'm very acquainted with spivak's calc on manifolds, it's a very...
28. ### Any good books on PDEs?

Taylor is extremely mathy. It does PDE's directly on manifolds, so I recommend to know a bit of differential geometry beforehand. So don't be surprised if you find Taylor too much to handle.
29. ### Any good books on PDEs?

A good intro book seems to be Strauss: https://www.amazon.com/dp/0470054565/?tag=pfamazon01-20&tag=pfamazon01-20 More rigorous books (and not exactly meant for a first course) are Evans and Renardy: https://www.amazon.com/dp/0821849743/?tag=pfamazon01-20&tag=pfamazon01-20...