/sci/ - Science & Math

Oh, and if you've got any questions, not necessarily related to the proofs given in the images, feel free to ask.
Also, I'll happily correct my files if you find any errors.

2/5

Virus

>Virus
If maths is causing sleeping problems to you, then yes.

3/5

So a collection of questions asked by the retards that frequent /sci/?

4/5

Josef, are you a mathematician by profession?

>>1325188
That's a pretty rude way of saying it.

5/5

>>1325197
There are tons of underage summer faggots in here today. Just ignore them.

>>1325195
Nope, I'm studying physics, but there's a lot of maths involved.
(Those proofs are really very basic, you learn most of the stuff in the first semesters)

>>1325207

The sin(x)/x proofs are beautiful. Thank you very much indeed.

JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD JOSEF FOR MOD

>>1325214
I just added them because of the thread an hour ago.

>>1325215
There are no mods here! Stop trying to break the laws of nature. :C

>>1325215

Mods are global, specific boards have janitors.

Josef for janitor :P

>>1325231

I have always enjoyed your posts as insightful Josef. In light of the spamming raid going on right now, I would certainly nominate you for janitor.

>>1325244

Well he has my vote.

Stop hijacking my thread with kissing my ass guys! ;(

>>1325256

Sorry I'm used to the trip-friend circle-hugs on /tv/.

>>1325172
For the proof of limit as x tends to 0 of sin(x)/x = 1

Isn't it wrong to use L'Hopitals rule for it?
As in order to differentiate Sinx from first principles you have to prove that the limit as x tends to 0 of sin(x)/x = 1 anyway. So using that result in a proof of it seems a bit sketchy.

>>1325263
Could be, could not be. I'll have a look at it, hold on.

>>1325244
I second this, we need more people like him.
Although I don't always understand everything he puts.. haven't started university yet.

>>1325263
Looks like you're right, I've corrected it by adding "Note that there's another ``proof'' around, which is wrong because of circular reasoning: ---- This proof uses \ieq{\tdq{}{x}\sin x = \cos x}, but in order to prove that, you already need the limit of \ieq{\sin(x)/x}."
I'm not sure whether it's smart to write something wrong in a collection of proofs though.

Anyway, thanks for clearing that up, I didn't know that yet.

>>1325327
No problem, it can be quite deceptive.

>>1325327
Which university do you study at Josef?

please STOP POSTING

>>1325379
Würzburg, Germany.
It's a bit different in Germany, we don't have such a thing as elite universities. It doesn't matter much where you study, unlike in the USA.

>>1325381

lol u mad?

i agree, no one should be allowed to post math on /troll/

>>1325391
I'm in the UK, we definitely do have elite universities.
Is the rest of Europe like Germany in that sense or are you guys an exception?

>>1325401
HAHA DISREGARD THAT a single post is worth a thousand sages

OMG SOMEONE ON THE INTERNET KNOWS MORE MATH THAN ME

SAGE

>>1325406
I actually don't know. I mean I've heard of a few elite universities in other countries, e.g. sorbonne, but that's about it.
Apart from that, I don't think a university could be much better than mine, even the elite ones. Maybe more teaching assistants. My boss has beam time in Berkley on a regular basis and he says "everything that's different about an elite university is that nobody asks what's different at an elite universits there". :)

>>1325406
In Norway we have two or three "elite" Universities..
But all of the rest is "good" - Don't have any bad ones..

The good ones are:
NTNU in Trondheim (the best).
University of Oslo
University of Bergen

>>1325424
Oh and before you google it, my university has shit ratings. Reputation used to be pretty good, but then they were forced to introduce the bachelor/master system, and everything went shitty. Nobody here knows how to organize shit anymore, and don't you dare having a question haha
The physics didn't change though

why do you write integrals like
<div class="math"> \int dx \; f(x) </div>
as opposed to
<div class="math"> \int f(x) \;dx </div>

>>1325426
sivingfag detected

NTNU is best at engineering and engineering only. Everything else is severely underprioritized and you'd (not you my friend, general you) be wise to move to Bergen or Oslo if you want to choose anything else than something engineering related. I mean, derp, the campus for LOL NOT TECHNOLOGY OR SEVERELY UNDERPRIORITIZED PURE SCIENCE is located in the middle of the forest or something. Been there once, almost got ravaged by a bear.

>>1325530
It is something physicists do to be more clear about what variable they are integrating over.

>>1325406
Pretty sure the rest of Europe is the same way, I can certainly vouch for the Netherlands.

>>1325558
i think that notation is absolutely retarded, but if he decides to use it at least he should be consistent

>>1325590
It is sometimes relieving if there is a lot of variables, like the case is for the fourier transform. But if there is only one variable there is no point.

>>1325172
Josef, did you use a Taylor Series and or LHopital's rule to show that sin(x)/x=1 as x goes to 0? Circular reasoning doesn't work because circular reasoning doesn't work.

Taking the derivative of the sine function requires that limit (and (1-cos(x))/x as x goes to 0.) You can't use LHopitals rule unless you somehow already knew that limit.
The taylor series also requires derivatives to be taken.

I expected more from you. :[

>>1325622
you can define sin by its series, and in fact this is the easiest way to define it on the entire complex plane

>>1325622


>You can't use LHopitals rule unless you somehow already knew that limit.

>>1325621
well im not sure how much difference it would have made by putting <span class="math"> dx [/spoiler] AFTER the function that is being integrated over

GTFO NAMEFAGS

>>1325639
just warning the reader so he won't think OH I SEE A LOT OF Ks I SUPPOSE WE WILL INTEGRATE OVER THEM and then the dx appears and he is all NOOOOOOO WHAT IS HAPPENING NOW I NEED TO READ IT ALL OVER AGAIN

derp I dunno, silly physicists, eh

sin(x)/x is pretty intuitive

1/x is veeery large x goes to 0

sin(x) is veeeeery small as x goes to 0

their product must be a real number

>>1325658
someone sucks at math

>>1325631
That's what I said. Take the derivative of the sine function using the limit definition. The limit he's trying to prove appears in the calculation.

>>1325630
I don't know how you find taylor series, but where I come from, I use this formula:
http://en.wikipedia.org/wiki/Taylor_series#Definition
There's a few cases where you don't need to rely on this formula (like in the case of a function being it's own derivative ((turns out to be the exponential function)) or any finite length polynomial) but this is the general formula.
inb4 sine is a linear combination of exponentials... the complex exponential also needs calculus (as far as I know) to be written as sin and cos... the sine taylor series isn't immediately obvious like the exponential one

>>1325652
ok i suppose that's a fair reason

Enough of the tripfaggotry.

>>1325630
>you can define sin by its series, and in fact this is the easiest way to define it on the entire complex plane
>easiest way

I also will disagree with this. I'm not sure what you mean by "easiest way," but I find myself writing it in terms of exponentials far more frequently than a polynomial. I only use the polynomial form occasionally while dealing with residues and the like.

>>1325666
the complex exponential is defined by a series, and we define <span class="math"> cos z = Re \exp(z) [/spoiler] and <span class="math"> \sin z = Im \exp(z) [/spoiler]. we can then express them in terms of the exponential. to find their series all we need to do is scale and add term by term (which is valid since the series for exp converges everywhere)

Looks like Josef already addressed my post
>>1325622
here:
>>1325327

>>1325679
my mistake i was thinking about the exponential form and then deriving its series. defining it in terms of the complex exponential is without a doubt the easiest way.

>>1325666
You can define sin x as its taylor series, pretend that's the starting point instead of LOL TRIANGLES and then prove everything you knew about sin x from the taylor series. I mean, alright, we found the taylor series with the previous definition, but we'll just pretend we didn't and start all over again with this series as the definition of sin x instead of an arbitrary definition.

>>1325682
Show, without taking the derivative of sine and cosine (no lhopitals rule or magically knowing the series expansions of sin and cos), that e^(ix)=cos(x)+i*sin(x). I submit that you are relying on circular reasoning when you define cos(x) as Re[e^ix].

>>1325676
No namefaggotry for the last forty minutes, mister!

>>1325708
i deleted my post because i realized i was doing something wrong, but i will address your point

for any real number <span class="math"> x [/spoiler], i DEFINE <span class="math"> \cos x [/spoiler] to be the real part of <span class="math"> e^{ix} [/spoiler] and then i DEFINE <span class="math"> \sin x [/spoiler] to be the imaginary part of <span class="math"> e^{ix} [/spoiler]. euler's formula then immediately follows.

>>1325700
I'm interested in this.
Define f(x) as
<div class="math">f(x)=\sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!}</div>

Now, show that f(x) is equivalent to sin(x) in the LOL TRIANGLES sense. Ie, the ratio of two legs.

>>1325728
furthermore we extend the definition to the entire complex plane by setting
<div class="math"> \cos z = \frac{e^{iz} + e^{-iz}}{2} </div>
and
<div class="math"> \sin z = \frac{e^{iz} - e^{-iz}}{2i} </div>

from this it's clear that these functions are holomorphic and we can find its power series using termwise addition

>>1325728
That's great, but you haven't demonstrated that those are the regular sine and cosine functions. As far as I'm concerned, you merely redefined a commonly used symbol. I know you're right, but you aren't being mathematically rigorous, which is necessary for proofs.

See >>1325752

Found a nice proof in an elementary calculus book;
consider the picture on the left;
the small triangle 0,(1,0),(cos x, sin x) has area
(sin x)/2,

the big triangle has area
(sin x)/(2cos x).

The area in the middle (the diskpart) has area x/2.
So we now get for x close to 0;

sin x /2 < x / 2 < sin x / (2cos x) =>
1 < x / sin x < 1 / cos x =>
1 > sin x / x > cos x;

now use the squeeze(pinch) theorem to obtain desired result.

>>1325754

See:
>>1325762
and
>>1325752

>>1325769
Aaaand that's the wrong picture, my bad.

>>1325752
I'm a physicist, so I always just assume someone else has proved all those pesky little details.

But now I am curious. How would we go about proving something like this? It is pretty essential and important for everything to be okay, so someone must have done it and it must be possible.

>>1325762
this is a standard result from complex analysis

suppose <span class="math"> f [/spoiler] and <span class="math"> g [/spoiler] are functions holomorphic on an open connected subset <span class="math"> U [/spoiler] of the complex plane such that there is a sequence <span class="math"> z_n [/spoiler] in <span class="math"> U [/spoiler] where <span class="math"> f(z_n) = g(z_n) [/spoiler], and <span class="math"> z_n [/spoiler] converge to some <span class="math"> z [/spoiler] in <span class="math"> U [/spoiler]. then in fact <span class="math"> f = g [/spoiler] on <span class="math"> U [/spoiler].

Butthurt antitaylor-series samefag here

>>1325775
The proof I know is very similar to this. This is acceptable for sin(x)/x as x=0

>>1325769

No geometrical proofs.

Josef is the king of /sci/

>>1325792
I'm an engineer. You should know where your math comes from.

I can show that my f(x) is actually sin(x), but only from the starting point of sin(x) being the ratio of two legs. I don't know how to work backwards rigorously.
>>1325775
>>1325769
This proof can be used to find the derivative of sine (and a similar proof for cosine). Once you have those, you can find the Taylor series for both trig functions and easily relate them to the function that is it's own derivative (turns out to be the exponential after some more sexy proofs).

>>1325806
>Trigonometry
>No geometrical proofs.
gtfo

>>1325833

Trigonometry is glorified geometry, faggot.

>>1325841
I know that trig is geometry. I was attempting to point out how ridiculous it is for one to demand a non-geometric proof for a geometric concept. Read what I was responding to.

heres is your geometrical proof for sin(x)

animooted

>>1325854

I wish to see an algebraic proof. The geometric proof is very elegant, but algebra is more my thing.

>>1325875
The problem with that is that you'll have to proof it with almost no knowledge of sin x, apart from it being continuous, you can't even use that it's C^1. I don't know if it's possible, don't think I ever saw a proof.

>>1325875
>sine function
>algebraic proof, no geometry or calculus
good luck with that

>>1325530 Why do you write integrals like INT dx f(x) instead of INT f(x) dx
It's sometimes more convenient to have the dx right after the integral sign. If you're doing multiple integrals, you know what the integral is with respect to. It's just a notational difference.

>>1325622 Both sinx/x proofs are wrong
The first definition of sin I've ever seen was the series. It was more like a coincidence that the Taylor series is just the same thing.

>>1325825
Well, I was curious about the backwards direction since that's the kind of details I assume is alright without investigating further. Knew about the forwards direction. Still, it might be enough, but if someone were to establish this purely for the sake of proving sinx/x -> 1 as x -> 0 in a non-geometric way it would be a rather pointless exercise since they would need the geometric proof anyway to prove that the taylor series provides the same sin x, if there is no other way. HILARIOUS, isn't it. I hope someone has done this.

>>1325924
see
>>1325752

>The first definition of sin I've ever seen was the series. It was more like a coincidence that the Taylor series is just the same thing.
http://en.wikipedia.org/wiki/Trigonometric_functions#Series_definitions
Just deriving the series requires a geometric interpretation as far as I can tell. It doesn't seem like "here's an interesting-looking series... oh by the way, it just so happens to output the ratio of triangle legs" is very coherent reasoning.
My schooling started with a geometric definition, and then calculus was used to understand more properties and extend it to the complex plane.

>>1325976
I've never really needed the triangle version of sin after the 1st semester. Also, I've never understood how the function relates to triangles. I guess it's just a leftover of the definition of an angle in basic linear algebra.

>>1325996
>

>>1326021
Usually what you've done with triangles before can be done with rotational matrices. I mean that's actually what sines are all about, changing the direction of a vector without altering its length. At least in trig.

>>1326119
The geometric definition still seems necessary for that... unless you can manage to derive the rotation matrices from the taylor series for sine and cosine. (And again, using the exponential function is cheating unless you can show it's geometric interpretation without circular reasoning.)

Can you or can you not show that the series for sin(x) is the ratio between the legs of a triangle without starting from the geometric definition of sine?
I know it is true, but I really want to see it worked out backwards: ie starting from the taylor series and proving that it's the geometric function we all know and love... rather than starting from the geometric interpretation and deriving the series.

>>1327772
I can not, but someone else sure can.

>>1327772
in rudin's real and complex analysis he starts with the series definition for exp, then defines sin and cos via euler's formula. he then proves that <span class="math"> z \mapsto e^{iz} [/spoiler] sends the real axis onto the unit circle, and defines <span class="math"> \pi [/spoiler] to be the least positive real number satisfying <span class="math"> \cos \pi/2 = 0 [/spoiler]. from this we can recover the geometry of the trigonometric functions.

Christ, they were right.

Statistics is really the highest math normal people need. I feel like less of a man for being unable to understand any of this.

I want to go back to school for math. :o

>>1327879
>>1327879
>>1327879

THIS is what I'm looking for. Is there an online print of this proof somewhere? I'd like to see it out of curiosity.

>>1327971
In particular, I'm interested in the proof for
>he then proves that z -> e^(iz) sends the real axis onto the unit circle

>>1327890
>Christ, they were right.

>>1327977

>>1327981

>>1327976
<div class="math">|e^{iz}| = (e^{iz})^*e^{iz} = e^{-iz}e^{iz} = e^0 = 1</div>
If you want the book, I know a really cheap online shop that sends them even at night.

>>1327998
... for real z, of course.
Apart from that, I don't really think that book is what you're looking for. It's about math, not triangles.

>>1327998
you also need to show that every complex number on the unit circle can be written as <span class="math"> e^{it} [/spoiler] for some real <span class="math"> t [/spoiler], which needs a bit of technical work

>>1327998
also you forgot to square the absolute value

>>1327998
>>1327998
Oh, derp... This is obvious. Excuse my stupidity on that one.

Now I'm interested to see that an argument in radians will map to the unit circle in positive orientation.
Specifically:
> and defines pi to be the least positive real number satisfying cos pi/2=0

After all, |exp(i*k*t)|=1 for all real numbers k and all real angles t. What makes k=1 so special aside from the fact that differentiating exp(ikt) cycles through the same four functions?

Essentially, I want to know why exp(i * pi/2)=0+1i
Something tells me that this is going to be as trivial as mapping real z to the unit circle but I'm not seeing it immediately. I excitedly await a response.

nice effort.

a short section about the residue theorem would be cool and very handy for newcomers, altough the actual proof is pretty boring. residue theorem really gave me the "shit brix" moment.

going to DESY tomorrow, ILC HCAL meeting. if i behave well my boss will let me fly to the main meeting in casablance in september, yay :D

>>1328035
>Essentially, I want to know why exp(i * pi/2)=0+1i

by definition of <span class="math"> \pi [/spoiler] and euler's formula we have
<div class="math"> e^{i\pi/2} = \cos \pi/2 + i \sin \pi/2 = i \sin \pi/2 </div>
since we already know it has unit norm, we must have <span class="math"> \sin\pi/2 = 1 [/spoiler] and hence <span class="math"> e^{i\pi/2} = 1 [/spoiler]

>>1328056
of course when i said 1 i meant i

Before I'm typing/thinking this stuff all by myself again I'll just copy the few pages and post them. pfff
1/4

2/4

3/4

4/4

>>1328052 residue theorem really gave me the "shit brix" moment.
Same for me :D
Have fun at the DESY, I envy you, got a shitload of UPS data to work with the next weeks

>>1328052
So ILC isn't dead? Or is it more alive in Europe than the US?

>>1328083

not at all.
no one knows when (if ever) there will be funding, but detector development is done with full power.
there are many new detector concepts for the ilc and we are trying out LOTS of things.
you just have to start really really early with the development process in order to have a fleshed out working and failsafe system once the money is availible to build the whole thing ;-)

i should add: the ilc will definitely not be built within the next 20years minimum.

>>1328061
>>1328062
>>1328063
>>1328071

<3

>>1328161
I always love new ways of thinking about these things.

Unfortunately, whenever I see limits I think of that scene from Mean Girls where Lindsay Lohan says "The limit does not exist!"

>>1328170
Glad I don't know what you are talking about. I don't want Lindsay Lohan (is she spelled like that? too lazy to scroll down) in my math thinking.

>>1328180
I'm not going to post the link to the vid because you're a good person who does not deserve to be corrupted like me ;_;

>>1328187
That kinda made me smile now. Looks like a good time to go to bed, I'm way past when I should've gone to sleep.
This thread is now open to trolls. ;)

Thread is still alive, wtf
Bumping for great glory!

A real scientifc thread, on /sci/, WHAT IN THE MOTHER OF GOD HAVE YOU DONE JOSEF?

>>1330442
Sorry, I didn't intend to sci /troll/

Josef is now awarded the prestigious Kimiko Ross Medal (Knight of /sci/)

Just dropped in to say thank you, Josef.
Also, if you want an easy way to prove <span class="math">\lim_{x\to0}\frac{\sin(x)}{x} = 1[/spoiler] is to define sin and cos as the only solutions f, g of these coupled differential equations :

<div class="math">\left\{\begin{align}
f' - g &= 0 \\
g' + f &= 0
\end{align}\right.,

\left\{\begin{align}
f(0) &= 0 \\
g(0) &= 1
\end{align}\right.</div>

Then it is clear that <span class="math">\frac{\sin(x)}{x} = \frac{\sin(x)-\sin(0)}{x-0}[/spoiler] is the difference quotient of sin about 0, so it's limit is by definition the derivative of sin at 0 : <span class="math">\lim_{x\to0}\frac{\sin(x)}{x} = \sin'(0) = f'(0) = g(0) = 1[/spoiler].

Thank you, Josef.

Also, if you want an easy way to prove <span class="math">\displaystyle\lim_{x\to0}\frac{\sin(x)}{x} = 1[/spoiler] is to define sin and cos as the only solutions f, g of these coupled differential equations :

<div class="math">\begin{cases}
f' - g &= 0 \\
g' + f &= 0
\end{cases},
\begin{cases}
f(0) &= 0 \\
g(0) &= 1
\end{cases}</div>

Then it is clear that <span class="math">\displaystyle\frac{\sin(x)}{x} = \frac{\sin(x)-\sin(0)}{x-0}[/spoiler] is the difference quotient of sin about 0, so it's limit is by definition the derivative of sin at 0 : <span class="math">\displaystyle\lim_{x\to0}\frac{\sin(x)}{x} = \sin'(0) = f'(0) = g(0) = 1[/spoiler].

>>1325172
This thread is good and you should feel good.

Thanks for the PDF btw, my university never even mentions convolutions since, for whatever reason, Fourier mathematics is skipped over. I know there is a course coming up that is Fourier-heavy though so a get-ahead is always nice.

Seriously, sticky this shit. It's the only useful thing /sci/ has ever done.

>>1330919
I should warn you that the delta and Fourier proofs aren't as rigorous as they should be in pure mathematics. This is a physicist's view.

Keep it up Josef, you are scaring 14 year old highschoolers with this glorious math

>>1325391
>It's a bit different in Germany, we don't have such a thing as elite universities.
your forgot about Bonn, did you not?

>>1325172
nice work, josef

>>1330946
I doesn't matter, I am a physics student myself. Rigour is only important to the physical level.

This thread: my mind blown

  ▲
▲ ▲

      ▲
    ▲ ▲

Thank you, Josef.

Also, if you want an easy way to prove <span class="math">\displaystyle\lim_{x\to0}\frac{\sin(x)}{x} = 1[/spoiler] is to define sin and cos as the only solutions f, g of these coupled differential equations :
<div class="math">f' - g = 0, g' + f = 0</div><div class="math">f(0) = 0, g(0) = 1</div>Then it is clear that <span class="math">\displaystyle\frac{\sin(x)}{x} = \frac{\sin(x)-\sin(0)}{x-0}[/spoiler] is the difference quotient of sin about 0, so it's limit is by definition the derivative of sin at 0 : <span class="math">\displaystyle\lim_{x\to0}\frac{\sin(x)}{x} = \sin'(0) = f'(0) = g(0) = 1[/spoiler].

>>1331014
Waaaait ... oh wow, I like that one.

>>1330980
Rigor is pretty important if you're trying to derive something new. The stuff I've posted here is already well-understood in mathematics, and the simple/unclean notation has been observed as working. However, I doubt that one could have built the whole formalism up with just those sloppy proofs.

And thanks for all the flowers by the way :)

G_\mu_ \nu +\Lambda_ \mu_ \nu =\frac{8\pi G}{c^4}T_\mu_\nu

ami doin it right?

>>1331089
No. The Einstein field equations are
<div class="math">G_{\mu\nu}+\Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu}</div>
There is no cosmological tensor, it's the cosmological constant simply multiplied with the metric

and you still forgot Bonn :(

>>1331098
I've never heard that Bonn would be some elite university, but I'll happily send my greetings up there if you're a student at that university. :)

>>1331095
Woops silly me I left out the g

How do I get it to actually show up? All I see is the code

I have a maths degree and I have no idea what's going on in this thread.

>>1331105
unfortunately I am not. But in Bonn there is the Max Planck Institut, the Hausdorff Center for Mathematics, the Bethe Center for Theoretical Physics, and the University of Bonn itself is quite good, too. (I like Teichner <3, and I believe M. Kontsevich graduated at Bonn, also Faltings, etc.)

Dear Josef @home

It is with great honor that we, in accordance with the established will of Alfred Nobel, hereby award you the Nobel Prize in Physics for your outstanding contributions to the field of teaching retarded /sci/-dwellers some basic mathematics. Congratulations on your momentous achievement.

>>1331107
• Use TeX/jsMath with the <span class="math"> (inline) and <div class="math"> (block) tags. Double-click equations to view the source.[/spoiler]</div>

>>1331143
Do you need to download TeX/jsMath? If so, how?

>>1331143
you are doing it WRONG
• Use TeX/jsMath with the [ math] (inline) and [ eqn] (block) tags. Double-click equations to view the source.

>>1331143
What are the (inline)and(block)tags?

Like this?
[G_\mu_ \nu +\Lambda g _ \mu_ \nu =\frac{8\pi G}{c^4}T_\mu_\nu]

>>1331014

You can't just define some functions as the only solutions to those DEs, you need a uniqueness proof to make that argument water tight.

Or like this?

<span class="math"> G_\mu_ \nu +\Lambda g _ \mu_ \nu =\frac{8\pi G}{c^4}T_\mu_\nu[/spoiler]

>>1331152
[ math] for equations in text (like $ in <span class="math">\LaTeX[/spoiler]) and [ eqn] for awesome equations (like \begin{equation} in <span class="math">\LaTeX[/spoiler]). Just type in [ math] e^{i\pi} = -1 [ /math] and get <span class="math"> e^{i\pi} = -1 [/spoiler]

I never downloaded jsMath and it works.

>>1325172
Your typesetting is inconsistent and overly traditional. Other than that, they're quite good.

I study physics as well.

Maybe like this
G_{\mu \nu} +\Lambda g_{ \mu \nu} =\frac{8\pi G}{c^4}T_{\mu \nu}

>>1331163
moar liek dis, but with correct syntax..

ACTUALLY LIKE THIS: <div class="math"> G_{\mu \nu} +\Lambda g_{ \mu \nu} =\frac{8\pi G}{c^4}T_{\mu \nu} </div>

<div class="math"> G_{\mu \nu} +\Lambda g_{ \mu \nu} =\frac{8\pi G}{c^4}T_{\mu \nu}</div>

>>1331167
>inconsistent

Yeah I noticed this. Use of italics is a little funny. Still very useful though, and one of the better things to come to /sci/ since its creation.

>>1331176
I DID IT YES OMG YES I DID IT

>>1331123

Faltings graduated and PhD'd in Münster.

lrn2namedrop.

*ahem* http://www.codecogs.com/components/equationeditor/equationeditor.php ?

>>1331177
Tip for readability: always use parenthesis with functions and operators.

>>1331179
oh shit!! take J.Jost + Kreck instead.
By the way: why the fuck is Kreck director of the MPM? Isn't he just a retard?

heh i root men
<div class="math"> \sqrt{-men}=i\sqrt{men}</div>

>>1331192

Well, they had to find SOMEONE to follow Hirzebruch, hadn't they?

>>1331195
but Kreck is really .. oh well.

>>1331195
Atiyah would have been the better choice.

>>1331220

Agreed. But Good doesn't always triumph.

>>1331228
>>1331220
>>1331213

Second thought... I don't know about that Kreck chap. So I better keep my gap shut. ^^

>>1331177
In what context?

(Oh, and italics for theorems are default in the theorem Latex package I'm using.)

Hi Josef. In case you are still around: http://scichan.org/nsci/res/5.html#171

>>1331587
Ah thanks, I assume that's some nice book about basic gauge theory?

>>1331629
propably not, but I do not know exactly what you mean with gauge theory.In any case it treats mostly the mathematical foundations of gauge theories (i.e. principal bundles, principal connections on them, and much more), but there is little physics in it. (Actually none, except for the last chapter on Yang-Mills theory.)

Have a look on your own: http://rapidshare.com/files/404927202/BaumEichtheorie.pdf

>>1331162
Yes, I was just suggesting that you can, in fact, use the differential equations to define sin and cos because the solutions are unique, but the proof was not really relevant.
What it amounts to is basically decoupling the equations to show that f and g are basically simple harmonic oscillators, that is <span class="math">f'' + f = 0[/spoiler] and <span class="math">g'' + g = 0[/spoiler].
From then on if you just have the basic theory of second order linear differential equations, you're pretty much done since the coupled equations give f(0),f(0),g(0) and g'(0) which assure both existence and unicity of the solutions.

Heck, you could probably use some variation of Cauchy-Lipschitz if you view the system as a second-order matrix differential equation (not possible to do matrices on jsMath here so bear with me) :

f' - g = 0
f + g' = 0

f' = 0*f + 1*g
g' = -1*f + 0*g

(f)' (0 1)(f)
(g) = (-1 0)(g)

AX = X'
(0 1) (f)
where A = (-1 0) and X =(g)

which has the unique solution X(t) = exp(-tA)X(0), given X(0), that is, given f(0) and g(0).

>>1331162
Yes, I was just suggesting that you can, in fact, use the differential equations to define sin and cos because the solutions are unique, but the proof was not really relevant.
What it amounts to is basically decoupling the equations to show that f and g are basically simple harmonic oscillators, that is <span class="math">f'' + f = 0[/spoiler] and <span class="math">g'' + g = 0[/spoiler].
From then on if you just have the basic theory of second order linear differential equations, you're pretty much done since the coupled equations give f(0),f(0),g(0) and g'(0) which assure both existence and unicity of the solutions.

Heck, you could probably use some variation of Cauchy-Lipschitz if you view the system as a second-order matrix differential equation (not possible to do matrices on jsMath here so bear with me) :

f' - g = 0
f + g' = 0

f' = 0*f + 1*g
g' = -1*f + 0*g

(f)' = (0 1)(f)
(g) = (-1 0)(g)

X' = AX

Where X is the column vector (f g) and A is the previous 2x2 matrix.
X' = AX has the unique solution X(t) = exp(-tA)X(0), given X(0), that is, given f(0) and g(0).

>>1325207

I'll learn this? Sweet. Going physics this wintersemester for the first time.

>>1331123

There are several Max Planck institutes

>>1332006
Complex analysis was in the 4th semester for me. Not sure about other universities, but it might take some time. It's not too hard for self-study though if you know some integration basics and if you're really interested.

>>1332032

In which semester are you now?

Can you give me examples of complex analysis applied to physics?

>>1332069 Can you give me examples of complex analysis applied to physics?
Uaaah. It's everywhere. See the proof doc for example: in order to find the Fourier transformation of a Lorentz curve, you have to know complex analysis. The residue theorem enables you to evaluate real integrals very elegantly that would be very hard or even unsolvable using classical integration. (It's of course handy for other purposes)
I've recently needed the Kramers-Kronig relation. I assume you know what an index of refraction is. You can extend this to a complex index of refraction; the real part is what you know as the classical index, the imaginary part tells you how much light is absorbed. In an experiment it is easy to determine the imaginary part by measuring the absorption; the real part isn't as friendly. Complex analysis now allows you to determine the real out of the imaginary part, that is, you can determine the classical index of refraction by measurement of the absorption of light in the crystal.

herp derp look at me im a namefag

>>1332024
yes, but only one for mathematics. Right?

>You can extend this to a complex index of refraction; the real part is what you know as the classical index, the imaginary part tells you how much light is absorbed

I wanted to hear this type of awesomeness. thanks.

The proof of Hirzebruch's signature theorem also uses complex analysis :(

>>1332117

Oh well

>>1332115
>butthurt failtrip

Everybody knows Josef's showing off, nobody cares because he delivers.

This thread has been alive for over 24 hours now. Pop the champagne.
Okay, I think it's about time to let it die now.