Mathematics

6:00 PM

For the theorem on existence and uniqueness of solutions of differential equations, see Picard's existence theorem. In complex analysis, Picard's great theorem and Picard's little theorem are related theorems about the range of an analytic function. They are named after Émile Picard. == The theorems == Little Picard Theorem: If a function f : C → C is entire and non-constant, then the set of values that f(z) assumes is either the whole complex plane or the plane minus a single point. Sketch of Proof: Picard's original proof was based on properties of the modular lambda function, usually denoted...

Daminark

"Theorem: $\ldots$ lots and lots of stuff $\ldots$"

Simply Beautiful Art

@BalarkaSen

Semiclassical

Which is true. That's easier to see if you don't use commas, I think: $\epsilon_{ijk}\partial_i A_j$.

Balarka Sen

@Simply I know the theorem. That's why I quoted it.

Alessandro Codenotti

The professor stated without proof the Casorati-Weierstrass theorem in class, which is the baby version of Picard apparently

Balarka Sen

6:00 PM

@Daminark I like that parsing :D

Simply Beautiful Art

@BalarkaSen :P

Balarka Sen

Yeah, C-W says the image of the punctured ball is dense in C

I think

Lozansky

Ah, I'm used to seeing it as $\epsilon_{ijk} A_j \partial_k$

Semiclassical

Similarly, for the other term you have $(\epsilon_{ijk}\partial_i B_k)A_j$.

Lozansky

But sure, relabeling gives the same result

Daminark

6:01 PM

@Alessandro Is it the baby version of Little Picard?

Or is it little Picard itself?

Balarka Sen

baby version of baby picard

Manolis Lyviakis

guys i need help

Semiclassical

If you then swap the labels of $j,k$ (which you can---dummy indices) you get $(\epsilon_{ikj}\partial_i B_j)A_k=-(\epsilon_{ijk}\partial_i B_j)A_k$ where I've used the antisymmetry of $\epsilon_{ijk}$.

Lozansky

Where we can swap i with any index

Manolis Lyviakis

any1 with knowlgde in complex analysis?

Balarka Sen

6:02 PM

this is the strong picard tho

Alessandro Codenotti

@BalarkaSen yup

Semiclassical

Which is just $-(\nabla\times \mathbf{B})\cdot\mathbf{A}$.

Yes of course

That work for you?

Totally

6:04 PM

mmkay.

This is one of the nice cases, as you might've noticed.

Lozansky

I'm actually doing an exercise I think you would recognize

Semiclassical

There's a few product rules in vector calculus which have 4 terms rather than 2 and are basically a huge pain.

Balarka Sen

@Daminark My favorite statement of Stokes' theorem is that if you draw a loop there's a lot of swirly in it

Lozansky

Yeah, gradient of dot product comes to mind

Daminark

SMBC?

Manolis Lyviakis

6:05 PM

0

Q: Prove that exist $f_1$ and $f_2$ such that $f=f_1+f_2$

Suppose $f$ is analytic on the ring $D(0,1;2)$ Through the Laurent series of $f$ prove that there exist $f_1$ analytic on $D(0;2)$ and $f_2$ analytic on the ring $D(0;1,+\infty)$ such that $f=f_1+f_2$ on the initial ring $D(0,1;2)$ . I dont know even how to start .I know that having the lauren...

Balarka Sen

Yep.

Semiclassical

@BalarkaSen The circulation of a magnetic field around a closed loop is directly proportional to the current enclosed by the loop.

Manolis Lyviakis

can you help me?

Semiclassical

I'm going to make a wild guess and say you're doing either angular momentum or magnetic fields. @Lozansky

(Probably magnetic fields with $\mathbf{A}=\frac{1}{2}\mathbf{B}\times \mathbf{z}\implies \mathbf{B}=\nabla \times \mathbf{A}$ or whatever it is.)

Balarka Sen

@Semiclassica Right, that's the Gauss's lemma.

I like to say "sum of local swirly = global swirly"

Semiclassical

6:07 PM

heh.

I just like having a physical context to it.

Balarka Sen

local swirly being curl at a point, and global swirly being the integral of the field along the loop

Lozansky

@Semiclassical No, showing that the energy density $\mathcal{E}$ satisfies the continuity equation $\partial_t \mathcal{E}+ div \mathbf{P} =0$ where $\mathbf{P} = \dfrac{1}{\mu_0}\mathbf{E} \times \mathbf{B}$

Semiclassical

I think "Currents generate magnetic circulation" is a nice slogan.

Lozansky

Lol wait theres no capital epsilon?

Balarka Sen

yeah that's not bad

Semiclassical

6:08 PM

Do $\mathcal{E}$ (\mathcal{E})

Ah, yes, continuity equation for conservation of energy.

Lozansky

Yeah

Semiclassical

Not going to lie, conservation of energy in E&M is a pain.

Lozansky

It was trivial using that vector identity

Semiclassical

Though angular momentum is worse.

Lozansky

Well this was for a region free of charges and flows

Semiclassical

6:10 PM

yeah, that makes life easier

Lozansky

Oh no, got an exercise in curvilinear coordinates now

Semiclassical

Have fun :/

Lozansky

Hope this one isn't too bad :|

Semiclassical

Try to work using vector calc identities alone as much as you can

That has the advantage of being coordinate independent

Lozansky

Because they are invariant?

Semiclassical

6:12 PM

Well, yes. But also because it helps avoid actually writing out the coordinate forms

Lozansky

Well, I don't have much choice now

Semiclassical

Perhaps not. It'll depend on the problem.

Lozansky

Need to compute scale factors and the divergence in this coordinate system

Semiclassical

Ahh, yeah.

Which coordinate system?

Lozansky

I think I should recognize it

$\cases{x = e^{u+v} \sin(u-v) \\ y=e^{u+v} \cos(u-v) \\ z = w}$

Semiclassical

6:14 PM

Yeah, I've seen that.

Hmm

Lozansky

Don't remember where I've seen it though

Something elliptical maybe

Semiclassical

Yeah. Looks reminiscent of bipolar coordinates or elliptic coordinates

Lozansky

Bipolar coordinates is an unfortunate name

Semiclassical

It really is.

Daminark

Good lord

Semiclassical

6:17 PM

It might be a rotation of one of those two? But yeah.

aaanyway.

out for a bit.

Lozansky

Cya later!

Manolis Lyviakis

0

Q: Prove that exist $f_1$ and $f_2$ such that $f=f_1+f_2$

Suppose $f$ is analytic on the ring $D(0,1;2)$ Through the Laurent series of $f$ prove that there exist $f_1$ analytic on $D(0;2)$ and $f_2$ analytic on the ring $D(0;1,+\infty)$ such that $f=f_1+f_2$ on the initial ring $D(0,1;2)$ . I dont know even how to start .I know that having the lauren...

complex-analysis analysis power-series laurent-series

can you help me?

Abdullah UYU

6:37 PM

$A=\{1,2,3,4,5\dots ,48,49,50\}$. $K$ is a subset of K that have 5 elements. Let $f(X)$ is the function that gives the sum of the elements of $X$. How many integers $f(X)$'s range have?

Philip Cherian

Open question: does anyone know of any mathematica scripts that compute integrals using Feynman parametrization other than this one? I'm almost certain that they must exist, but I can't for the life of me find them anywhere on the internet...

Abdullah UYU

forget about last question

Vrouvrou

@Hippalectryon i don't understand what you do ?

Anonymous

@BalarkaSen Are Ted's lectures available at 720p ? I can't read what he is writing on the blackboard!

Semiclassical

6:55 PM

@Lozansky Playing around a bit, I think the lines of constant $u$ and constant $v$ are archimedian spirals (in polar coordinates, $r(\theta)=a+b\theta$) of opposite rotation. But my Google-fu hasn't found a coordinate system with those features yet.

Weeeird

Lozansky

@Semiclassical Huh okay. Maybe plotting in matlab/mathematica might help recognizing it?

Semiclassical

Well, that's how I figured out that they were spirals :)

Lozansky

Ahh, I see :P

Well the scale factors are $h_u=h_v = \sqrt{2}e^{u+v}$ and of course $h_w =1$

I've completely forgot in what kind of situation the arise though. Pretty sure I've seen them before

Balarka Sen

@blue Really? I could totally see what he was writing when I watched it.

Which video are you having trouble with?