Mathematics - 2017-09-25 (page 5 of 5)

@Semiclassical Right now, I think there is something seriously wrong with my brain itself, because I have been feeling so anxious the past few months.

Semiclassical

ugh, that sucks

0ßelö7

@Semiclassical Hi Phil,

We still hadn't really worked an example with that transformation, so I should probably delay the due date for this problem.

The bit that you still needed is that the circulation does not change---it is the same for both flows.

This is because

Gamma=Re int w dz = Re int Phi' dz

so that circulation is related to the jump in Phi as you go around a loop. The value of Phi at the corresponding points on the loops will be mapped to each other.

Doing the integral in the original plane would be complicated because of the branch cut, but it is easy in the image plane.

Jasper

9:27 PM

@0ßelö7 So you are Ryan?

0ßelö7

we used a different notation in class: $\Phi$ is the potential and $w=\Phi'$ is the complex velocity

@Jasper no

@Semiclassical I don't know how the integrals should be related

Semiclassical

@Jasper Do you have to deal with panic attacks? With me it's never as sudden/unexpected as an attack would be, but more like letting pressure grow in a sealed container until it bursts

0ßelö7

I think he's saying the integral is invariant under conformal transformations (?)

Semiclassical

@0ßelö7 under the map, yeah.

That certainly sounds right

Jasper

@Semiclassical No, I have no panic attacks.

@0ßelö7 So are you gonna learn French or not?

0ßelö7

9:33 PM

no

Semiclassical

Note that $w(z) dz = \Phi'(z)dz = d\Phi$

Jasper

I have no idea why people label French as the language of romance. Other languages can be romantic too.

ALannister

1

Q: Airplane vector problem - wind due east, plane northeast; need help drawing a picture

A wind is blowing straight east with a velocity of 25 mph and an aircraft is flying pointed North East (.e., at a $45^{\circ}$ if the $y$-axis points north and the $x$-axis points east) and records an airspeed of $130$ mph. I need to determine in which direction and how fast the airplane is actu...

Semiclassical

It helps that French is literally Roman-ce language

0ßelö7

@Semiclassical yeah but what does that have to do with the conformal trafo?

Jasper

9:35 PM

@ALannister I did these problems in high school. We just needed to draw a vector diagram.

Secret

Last night dream is a very crazy nightmare that took me a great effort to twist out a good ending from it. It involves me being bullied by one of the biggest bullies back in my high school, who has properties of an infinite dedekind finite set, thus making him very hard to catch and imprison because there exists no countably infinite descriptions on what he afraid of. In fact, he only afraid of one thing, and it is the pain in his ring finger

Gotta be glad that his real life self is not THAT crazy

ALannister

@jasper that's what I'm having trouble with in this problem

Semiclassical

Not much, beyond the fact that it's written in a way which is independent of z

0ßelö7

hmm

Leaky Nun

@Secret and I haven't slept

and our timezones are similar

0ßelö7

9:36 PM

ok so the vortex term is $$\frac{\Gamma}{2\pi i}\log z$$ for the disk flow

Semiclassical

But you should indeed be able to transfer it to an integral over the image plane

Jasper

@ALannister The question sounds ambiguous to me. I don't know which is the plane flying with wind and which without wind, lol.

0ßelö7

@Semiclassical Ok so when I do a contour integral for that dude I go up one sheet, right?

ALannister

Plane headed northeast, wind is blowing east

leys any i let p be the plane vector and r be the vector for the real speed

and w the wind vector

0ßelö7

@Semiclassical Hmm, well what he's saying is it should just be the difference in values between sheets, right?

Because $\Phi\propto \log z$ and we're integrating $d\Phi$

ALannister

9:39 PM

then woukd p and r share the same initial point and form a triangle that w Would be the third side of? @jasper

Secret

In the dream, that bully of mine likes to run me over with a bike, hit me etc. because he said I am stupid enough to do so. He cannot be imprison as it only took minutes for him to pick any lock and free himself, and teachers and authorities have trouble pinning him down because there is no countable description to describes what he fears

Semiclassical

Maybe. I'm on phone right now and can't talk too much

0ßelö7

@Semiclassical Ok, he delayed it anyway so it's not an issue I guess

I've got to finish this fortran and I'll probably think more on this later

Jasper

@ALannister When you say the plane is flying pointed northeast, you mean it is moving northeast under the influence of the wind?

0ßelö7

@Semiclassical actually the log branches are $2\pi n$ apart

it's because of the periodicity of the exponential

so go around once and you get a $2\pi$...wow that makes sense

ALannister

9:41 PM

No, that's the direction the person who's flying it is trying to point it @jasper

Semiclassical

Sure.

0ßelö7

because that's what integrating the derivative (1/z) gives

Semiclassical

Log is merciful in that regard

(By contrast, ProductLog has no such mercy)

0ßelö7

ProductLog?

ah, lambert function

Jasper

@ALannister If so, then you just need to add the two vectors to get the net resulting movement.

0ßelö7

9:44 PM

@Semiclassical Well that's a really clever way to do that awful integral :)

And I can argue physically that the other integral should vanish because it's symmetric (or use the same argument as for here)

ALannister

Okay good. That's what I did, so I got the right answer.

now, how do I find the angle it's actually flying at, @jasper?

Jasper

@ALannister That would be the angle of the vector that is the sum of the two vectors.

ALannister

Right, but how do I find that given the sum?

Jasper

@ALannister Have you drawn the triangle already?

ALannister

Yea

Jasper

9:49 PM

Since the wind blows east, it will be an arrow pointing to the right.

ALannister

The wind vector is an arrow pointing to the right

Jasper

Yeah, so if you add the northeast vector to it, the sum will point somewhere between northeast and east.

From that pic you get the angle, because you are using right to mean east.

ALannister

Yes, I can see what the angle is supposed to look like. But how do I calculate it it?

Jasper

Use trigonometry!

ALannister

With what though?

i don't know a

Jasper

9:54 PM

Let me see...

OK, you have two lengths and one angle between those two lengths, right?

Use the cosine rule to find the length of the sum.

Then use the sine rule to find the required angle.

Done.

@ALannister

ALannister

cosine rule, you mean cousins =adj/hyp?

thats cosine =adj/hype not cousins. Stupid autocorrect

Jasper

No, no.

ALannister

Then I don't know what you mean by cosine rule

Jasper

Law of cosines

In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states c 2 = a 2 + b 2 − 2 a b cos ⁡ γ , {\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos \gamma...

ALannister

you should have said law of cosines then

Jasper

9:58 PM

Law of sines

In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of a triangle (any shape) to the sines of its angles. According to the law, a sin ⁡ A = b sin ⁡ B = c sin ...

ALannister

Again, you should have said law of sines.

Leaky Nun

@ALannister same rule, different names

2

@ALannister same rule, different names

Jasper

LOL

ALannister

<-- does not know it called that way.

Leaky Nun

<-- does not care.

3

ALannister

10:00 PM

In any case, thank you for both helping and laughing at me.

@leakynun not you. You're being a Clint.

6

But in any case thanks. Bye.

Jasper

Clint Eastwood? I hate cowboy movies, lol.

Simply Beautiful Art

@Secret I wonder what my dreams are like sometimes

Secret

My dreams is a place where almost every maths concept are as physical as ordinary objects

2

In particular, ever since I investigate set theory, infinite sets started to appear in a myriad of forms in the dreams, similar to what happens when I first learnt about 4D space

It should not be long before a physical banach tarski paradox appeared in my dream when I return my investigation to the Choice universe (ZFC)

Jasper

10:37 PM

@Secret Your dreams are fascinating.

Faust

11:39 PM

@Abcd yeah of course i am =)

Give an example of a group G and a subset A such that, for all g in G
and a in A, $ gag^{−1}$
is in A, but A is not a subgroup of G

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$Mathematics$ Mathematics