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.
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...
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.
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...
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.
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...
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...
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...
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...
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...
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...
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...
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...
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...
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}##
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...
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...
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...
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...
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.
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...
If you're currently taking calc 2, then I feel that ballentine is not for you. Try at least to complete calculus, and take LA and differential equations. Then you'll be much more prepared for books like Ballentine.
Regardless, I think it's insane to study QM without knowing LA in the first...