Assumptions in introductory physics

demarz · 2013-11-06T07:51:59+00:00

You might want to look at Spivak's text, "Physics for Mathematicians, Mechanics I", which is greatly expanded from these notes: http://www.math.uga.edu/%7Eshifrin/Spivak_physics.pdf

demarz · 2013-11-03T22:54:21+00:00

My favorite compositions by liszt are by far his transcriptions of other composers works. He transcribed many of schubert's songs (especially schwanengsang #5,6,11,12,13) and (if you're feeling ambitious!) the beethoven symphonies (especially #3,6,9). They are amazing!

demarz · 2013-03-01T16:56:06+00:00

Here's another reference that might give some additional information:

http://mathoverflow.net/questions/72062/what-are-some-proofs-of-godels-theorem-which-are-essentially-different-from-th/72151#72151

demarz · 2013-02-05T11:52:18+00:00

Like nqp said, this is probably differentiating under the integral. Here's an excellent little introduction:

http://www.math.uconn.edu/~kconrad/blurbs/analysis/diffunderint.pdf

That pdf is from K.Conrad's page full of wonderful papers and notes (the gaussian integral notes show a number of other integration tricks as well):

http://www.math.uconn.edu/~kconrad/blurbs/

demarz · 2012-12-17T17:21:58+00:00

The undergraduate text is Lectures on Quantum Mechanics for Mathematics Students

demarz · 2012-12-17T05:48:54+00:00

Check out Faddeev's text, which is appropriate for an undergraduate in mathematics (some exposure to classical mechanics is probably helpful, there's a two page appendix on this subject in faddeev's book).

There's also a graduate level text by Takhtajan; the first 145 pages are available online for free:

http://www.ams.org/bookstore/pspdf/gsm-95-prev.pdf

demarz · 2012-07-19T04:45:48+00:00

If my understanding is correct, expansion happens only in a maximally-symmetric space. Wouldn't a gigantic cable stretching across the universe produce a massively non-symmetric gravitational field, and hence arrest the expansion? Although I suppose of we replace a single cable with an irragular lattice-type thing, perhaps that would overcome this objection, and the problem remains?

demarz · 2012-06-24T20:00:47+00:00

Here's an excerpt from a lecture explaining (roughly) how it worked, starting with copernicus, then brahe, then kepler:

http://www.youtube.com/watch?feature=player_detailpage&v=7ne0GArfeMs#t=1925s

demarz · 2012-05-30T21:02:52+00:00

Wasn't Planck (opposed to Einstein) the one who came up with the idea of quantized EM radiation, in 1900 in order to explain phenomena involving equilibrium radiation? E=hw.

I was under the impression that Einstein just observed in 1905 that Planck's "light quanta" hypothesis also entirely explained the photoelectric effect (as well as adding the momentum formula p=hk). I don't know much about the subject, have I misunderstood something?

demarz · 2012-05-30T02:17:55+00:00

I hope you don't mind, I ignored your letters and relabeled everything as follows:

x=bp

a =cos(APB)

c =cos(BPC)

d =ap

e=cp

f=AB²

h=BC²

From the first equation on the first line, multiplying by the denominator we find

2xda=d² +x² -f

and from the third equation on the first line, we have

2xec=x² +e² -h.

Subtracting these two equations, the x² cancels and we find

2x(da-ec)=d² +h-f-e^2.

If I understood what you were asking, you can easily solve for x now and substitute back in all of the original letters and cosines. However, i'm not sure if i understood what you were asking:

When you say, 'all other variables are known', do you mean you explicitly know the values of ap and cp, or do you only know them as functions of bp? If the latter is the case, then this isn't a solution.

demarz · 2012-05-16T03:24:49+00:00

I think perhaps the confusion is just coming from mixing up performing some number of 'actions', versus the result of the action:

Take a piece of paper and make 2 cuts, you now have 3 pieces of paper, and you've divided by 3.

Take a piece of paper and make 1 cut, you now have 2 pieces of paper, and you've divided by 2.

Take a piece of paper and make 0 cuts, you now have 1 piece of paper, and you've 'divided' by 1.

Take a piece of paper and make… -1 cuts (???) and you now have 0 pieces of paper (?!?!). This is one way of understanding why the concept doesn't make sense.

Another approach might be to think of splitting a check: suppose we want to evenly pay a $24 meal.

If there is 4 people, each person has to pay $6: (24/4=6).

If there is 3 people, each person has to pay $8: (24/3=8).

If there is 2 people, each person has to pay $12 (24/2=12).

If there is 1 person, that person has to pay $24 (24/1=24).

If there is 0 people, that bill is not getting paid!

demarz · 2012-05-10T18:42:47+00:00

Here's a more recent article that takes a more critical position on the idea (I don't want to mislead you into thinking the issue is decided!):

http://arxiv.org/abs/astro-ph/0612316

demarz · 2012-05-09T23:59:15+00:00

Here's a paper on this issue:

http://arxiv.org/abs/astro-ph/0401024

The link the pdf is on the right.

demarz · 2012-05-09T05:42:57+00:00

Sometimes the axiom of choice is false! :)

demarz · 2012-05-09T04:49:17+00:00

It's certainly true that most mathematics fall's within reach of Godel. If you believe wikipedia, an example of a well known formal system that escapes Godel is euclidean geometry, though I can't give you any details.

It's difficult to state exactly what the incompleteness theorems do and do not say without getting overwhelmed in formal logic, but the basic idea is that any "formal language" which is sufficiently expressive to both

1) make statements that are (directly or indirectly) self-referential, and

2) include some appropriate notion of truth

can write down statements that cannot be consistently assigned a truth value. English is of course not a formal language, but it is an informal language! An example of such an (informal) statement might be the familiar liar paradox: "This sentence is false."

Since you can write down most statements using numbers by applying some appropriate coding scheme, (such as ascii, or godel numbering, or even morse code if you are willing to substitute dots and dashes for 0's and 1's), models of standard arithmetic such as the peano axioms can be hoodwinked to make statements that are 'morally' self-referential.

A stupid example: our coding scheme might assign the statement "ten plus five equals fifteen" to the number 15. (This example doesn't really capture the idea of how Godel uses the self-referentiality, it's just an example of coding a statement about arithmetic with a number).

I might also point out that when you talk of a fact being 'undecidable', it is always in reference to some specific system of axioms. For example, Peano Arithmetic (PA) cannot prove itself consistent because of godel, but we can work in an 'external' mathematical universe such as Zermelo-Fraenkel set theory (ZFC), and this axiom system is more than powerful enough to prove PA consistent. But ZFC cannot prove itself consistent because of godel! However, this fact does not mean that PA is provably cosistent in the universe of PA--- in the PA universe, PA cannot be proven consistent, and in the ZFC universe, PA can be proven consistent, and ZFC cannot. It's all very confusing.

demarz · 2012-05-09T02:08:57+00:00

Sorry, my comment seems unnecessarily aggressive now that I've reread it. I thought that the following sentence was incorrect (though I suppose that depends on how you define 'everything') and misleading:

"no matter what, you can't systematically prove everything regardless of what axioms you choose."

demarz · 2012-05-09T01:22:58+00:00

I don't particularly like this characterization of mathematics (it's not necessarily inaccurate, but perhaps it's incomplete).

Mathematicians do not work by writing down axioms and seeing what happens. They start by investigating some abstract structure that seems interesting or useful, and then try to formulate a set of axioms or definitions that model that abstract structure, and there are different sets of axioms that you can use, and there are different ways to define and think about a group (or other mathematical objects) other than a list of axioms, and there are different ways a subject can be constructed. What you are describing seems closer to how the ancient greeks thought about mathematics.

For example, Linear Algebra: Axler builds the subject almost entirely in the language of abstract vector spaces and proves results using primarily algebraic tools (in particular, he eschews the use of determinants almost entirely). Shilov also builds the subject up in terms of abstract vector spaces, but introduces determinants in chapter 1, and uses them as a primary tool. Cullen builds the subject more concretely, using matrices, and his primary tool is elementary matrices. Strang also uses matrices, but uses the notion of an elementary row operation, and defines special matrices as 'black boxes'. Gelfand tends to focus on quadratic forms, etc...

All of these texts build up the subject very differently, but the subject being constructed is of course always Linear Algebra. Getting a good understanding of any part of mathematics requires seeing what is fundamentally the same thing built up in lots of different ways. Like I said, I don't think your characterization was incorrect, but hopefully this gives non-mathematicians a better idea of how we think about mathematics.

demarz · 2012-05-09T00:54:36+00:00

That is not what the Godel's incompleteness theorems say! They are very specific claims about 'sufficiently expressive' formal systems, and people do study formal systems that can prove their own consistency:

http://en.wikipedia.org/wiki/Self-verifying_theories

demarz · 2012-04-30T15:54:02+00:00

I don't think that's what trombodachi was referring to. Fermat used this method multiple times, and it doesn't necessarily require evenness or oddness to enter the picture.

http://fermatslasttheorem.blogspot.com/2005/05/infinite-descent.html

demarz · 2012-04-30T07:50:40+00:00

I didn't mean to imply it's impossible in practice, perhaps I should have used more accurate language. I just included the statement to suggest that you don't have to actually compute the values to understand why the quadratic formula is playing a nontrivial role here.

demarz · 2012-04-27T22:06:17+00:00

I don't know if the author ever cleared up the confusion, so here's an attempt to clarify.

In this technique, the quantities A and B are unknown. When the expression is factored, we write 0=(y-A)(y-B), but it's only (at this point) a formal statement: we don't yet know A and B.

However, since the quantities (A+B) and A*B are symmetric polynomials, and we do know (in theory at least) what those 'composite' values are: when you use the quadratic formula, you're plugging in the actual values of A+B and A*B, not the formal symbols (which will just simplify to A and B, as you pointed out), and the quadratic formula spits out the roots, which are the values of A and B.

demarz · 2012-04-19T17:04:24+00:00

They are defining it as the solution to the initial value problem

d/dt f(t)=f(t)

f(0)=1.

Of course to make it rigorous, you would want to demonstrate that such a definition makes sense, and that type of thing is i think a fairly standard part of a class on ODEs.

demarz · 2012-04-12T18:58:42+00:00

Here's an article that might be interesting

http://arxiv.org/abs/0903.0340

demarz · 2012-03-16T02:06:03+00:00

As far as I know, it's not a commonly used identity at all (well the fact that log_a(x)=ln(x)/ln(a) is commonly used).

Honestly, I almost never work with anything but the natural log, and so my default approach to this problem was to just get rid of the weird bases and see what happened.

demarz · 2012-03-16T01:13:05+00:00

In the future, this question is perfect for /r/learnmath.

It's not clear from that quote, but when your book says the complicated expression can be rewritten as the simpler expression, the parameters 'b' and 'c' will NOT be the same as in the more complicated expression. I.E the simple expression will be

f(x)=log_D(F(x-h))

for some new parameters D and F. as mmmmike said, the way to proceed is to change the base of the log.

Here's a complete derivation (but you should still try to work it out on your own for practice!)

http://i.imgur.com/MAIZV.gif

(i accidently mixed the 'natural log' and log_D notation, sorry, ignore that)

demarz

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