Mathematics - 2017-01-05 (page 1 of 4)

Akiva Weinberger

00:00

Sure, and then solve for y, and then plug that value into one of the equations to get x.

Or,

Find y in the first equation, plug it into the second, and then solve for x there

Remember you have tan(54*54')=(40+x)/y and tan(47*30')=x/y.

Sorry, typo, I meant 54*54 and not 54*40. Fixed.

MATH ASKER

Ohhh okay

now I get it I'll try to solve it thank you very much

Akiva Weinberger

So we have y=(40+x)/1.423, from the first equation, I think

or y=0.703(40+x) I suppose.

MATH ASKER

I got y as 118.8 when I did put x in the second equation

I did something wrong I guess I got y as 118.8 and x as 129.03

I did tan(47°30')= 1.428y-40/y, 1.0913y-1.428y=-40 y=118.8

Semiclassical

hi chat

Akiva Weinberger

@MATHASKER It's 1.4228, not 1.428.

You forgot a digit.

But besides for that, what you were doing should work, I think.

MATH ASKER

00:13

ohh so would x be 131.596

Oh ok thanks for helping

Akiva Weinberger

Yup!

So the tower without the antenna is 131, and the tower with the antenna is 131+40=171.

@Semiclassical Hyello

MATH ASKER

thanks a lot for the help man, right now I'm taking advance function and my teacher said that this was a pre calc level math

idk if i should go to pre calc lol I was planning to but i guess i need a lot practice

Secret

Category (mathematics)

In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. On the other hand, any monoid can be understood as a special sort of category (with a single object whose self-morphisms are represented by the elements of the monoid), and so can any preorder. In general, the objects and arrows may be abstract entities of...

Hmm, it does feel like categories are monoids where the underlying elements are the mophisms and objects

(except it is the other way around)

Zaid Alyafeai

Is this spamming proof ?

Secret

00:35

i.e. Define a + such that given any morphism $\eta$ it is a homomorphism wrt +

distributive law is a special case where $\eta$ is the set of multiplication by some element

Null

@Secret why do you post so much? :) Not saying it is uninteresting

Secret

because of giving a rundown or something? Unlike those zero term algebra posts, these posts is actually some analysis of papers. I know there are some people who are interested in them and thus I just left those posts there and if a discussion is sparked, there will be.

One thing I noticed is some times in the past there are many maths chat people discussing or quoting about wheels, but the details were not really fleshed out. This analysis serves to fill in that hole

well, of course there's also a simpler reason: It is quite boring to just stare at a paper without saying something, perhaps I am lonely or bored or something. The mere existence of these posts give a nonzero probability of someone responding to them and hence generating discussions

Null

01:07

@Secret fair enaugh!

Secret

01:34

These two rules still makes no sense to me. Why not (x+0y)z=xz+0yz?

Ramanujan

In 3D when 3 points which are collinear given,if I take determinant of those 3 points,i get 0,why?

Secret

Think what's the volume of the parallel pipped span by the vectors made by those 3 points

Ramanujan

0

Secret

and then you have your ans

Ramanujan

So points given are vectors [AB AC AD] ?

Secret

01:45

they are all vectors based on the origin (i.e. positional vectors)

Ramanujan

>in a plane there can be maximum 3 non collinear points.

What it means?

Secret

you can only have at most 3 noncolinear points. Think what happens when you try to introduce a 4th point and how its displacement vector will be represented on the plane

Fruitful Approach

What is the general term for a block of math text?

Say a definition theorem or proof

I'll just call it block of text :P

Kasmir Khaan

02:13

hi chat

Kasmir Khaan

02:46

hi @GFauxPas

I need your help with surface integral

GFauxPas

Hello Kasmir what makes you think I can help you?

but what's your question?

Kasmir Khaan

because you helped me before

F = ( x,y,z) / ( x^2+y^2)

i need to calculate the flux out of the cylinder Y : x^2+y^2 =2

-2<z<2

GFauxPas

Sorry friend I only know how to deal with conservative fields for these types of problems

Kasmir Khaan

its okay thanks anyway :)

GFauxPas

you'll have to go to the definition I suppose

Akiva Weinberger

02:53

Most fields seem to be conservative, I find;

GFauxPas

flux of what, water?

Akiva Weinberger

it's the urban centers that are liberal.

GFauxPas

rimshot

wait Kasmir what is this a flux of

water? electricity?

I suspect en.wikipedia.org/wiki/Divergence_theorem will help you @KasmirKhaan

look at this example: en.wikipedia.org/wiki/Divergence_theorem#Example

or khan academy probably

Semiclassical

03:12

Start with the definition of flux, and write down the appropriate integral. Then do that integral.

Kasmir Khaan

hi again

@GFauxPas @semi thanks i solved it now

@Semiclassical thanks i solved it now, the problem was that i didint know the physical meaning of the problem so didnt knew what I were calculating

Semiclassical

ah.

one setting would be an electric field, though that'd require a bit of an odd charge distribution.

Balarka Sen

I struggled for the last 10 minutes to put a picture I had (as a solution to a problem I was working on) into words until I realized it's just the mean value theorem...

Semiclassical

Actually, electric field wouldn't work so well since it's not curl-free.

But hey, don't need a literal physics case in order to apply the definition.

Akiva Weinberger

@BalarkaSen On a slightly philosophical level — the reason we have all this technical terminology in math is so we can explain our pictures

(I'm tired)

Balarka Sen

03:25

I agree

Semiclassical

With the philosophy, or the tired? :p

Balarka Sen

With the philosophy

Akiva Weinberger

He could agree or disagree with my tiredness; it wouldn't change anything

I need to do history. What are five historian opinions on the annexation of Texas?

Balarka Sen

i dunno man

but i do feel sorry for ya

Semiclassical

Could not tell you.

Akiva Weinberger

03:29

Maybe this chat would be better at answering "what are five mathematician opinions on the annexation of Texas"

Balarka Sen

most of them would be one-liners

Semiclassical

It was a really good thing, it was a mostly good thing, it was a mixture of good and bad, it was a mostly bad thing, it was a really bad thing.

There, five opinions.

Balarka Sen

also there aren't too many mathematicians around, but i think there are 5

Akiva Weinberger

@BalarkaSen What were you working on, anyway?

Balarka Sen

@AkivaWeinberger Counting critical points of a function on the circle. It was an easy exercise anyway.

Working through a bunch of differential topology exercises from Hirsch's big tome.

Have you went beyond derivatives yet?

Semiclassical

03:35

Apropos of mostly nothing, this is a neat picture.

(to get the mental motivation behind that, consider walking around one of those red circles and plotting the local direction of those lines.)

(If I walk around one of those red circles CW, the direction rotates 180 degrees CCW)

Balarka Sen

That's known as the index of the vector field at a zero, but this one's around a singularity.

Semiclassical

Right.

The subtlety here is that this is a so-called director field (physics terminology). point being that there's no sense of left as being different than right.

or, more simply, the map is from that plane to RP^2 rather than to S^1.

nematics are neat for that reason.

Balarka Sen

huh

@SemiC @Akiva Here's something neat; we all know the hairy ball theorem, which says there's no nowhere zero vector field on the sphere. There's an obvious vector field with two zeroes; is there one with exactly one zero? if so, what's the index around it?

I suppose you both know the picture tho

Semiclassical

I don't, actually.

Balarka Sen

stuff to think about, then!

Semiclassical

03:42

Heh.

Well, the obvious vector field with two zeroes would have lines of latitude being flow lines.

Balarka Sen

yep

Semiclassical

the difference between the two zeros being that the field winds around CW around one and CCW around the other.

Akiva Weinberger

@BalarkaSen Take a tangent line to the North Pole

and look at the set of planes that contain that line.

Semiclassical

if I imagine sliding both of them towards the equator, then I think I get something like a an ideal dipole field.

Balarka Sen

@Akiva boo

Semiclassical

03:46

except on the surface of the sphere.

Balarka Sen

you're right though.

@Semiclassical That sounds right to me

Akiva Weinberger

And it has to have index 2 no matter what, right?

Semiclassical

like this but projected onto the sphere.

Balarka Sen

Exactly right, @SemiC. it's what the picture Akiva's explaining: there's a 1-parameter family of circles passing through the north pole; now set up the vector field on each circle appropriately.

Akiva Weinberger

Alternatively, just take a constant map on the plane and stereographically project it to the sphere.

Semiclassical

03:47

@AkivaWeinberger I may try constructing that in Mathematica.

Balarka Sen

@AkivaWeinberger Correct.

Akiva Weinberger

(Cont'd) Which I think gives the exact same field.

Semiclassical

I think so too.

Balarka Sen

@Akiva you don't want the constant vector field. you have to dampen it towards infinity

otherwise you get singularity, not zero

but yeah, that sounds right

Akiva Weinberger

I'm imagining the constant vector field on the plane being illustrated as a bunch of arrows drawn on the plane. If we project that, the arrows on the sphere get smaller near the North Pole.

Balarka Sen

03:50

Oh right.

I realized after I said it.

Akiva Weinberger

Arright

I wonder what the one-singularity map looks like on other surfaces (like a genus-2 surface)

I can see a two-singularity map pretty easily. EDIT: Wait, no I can't

Balarka Sen

It's got to have index -2.

Akiva Weinberger

What's that look like? Six hyperbola branches or something?

Balarka Sen

It's just the same as the one you found on S^2, but all the arrows reversed.

(yours has index 2)

Akiva Weinberger

Oh

Balarka Sen

03:54

I don't immediately see how to do it on $\Sigma_2$, but the key is to draw the standard vector field smash togather the zeroes.

Akiva Weinberger

Wait, why can't I then just take the field we have on S^2 and reverse all the arrows to get a field with just a -2 pole

I think it has to actually look different

Aren't sources and sinks both +1

Balarka Sen

Yes, I am being silly. The index -1 thing is like (-y, x)

"saddle"

You win :P

So try to smash togather two index -1 things to see how an index -2 thing looks like maybe

Semiclassical

hmm, this is okay:

Akiva Weinberger

Is this gonna be a Mathematica illustration

Semiclassical

Akiva Weinberger

03:58

Yep

Balarka Sen

neat

Semiclassical

Not perfect, but it illustrates the point.

Akiva Weinberger

And then make all the vectors point along those circles.

Semiclassical

Sure.

arctic tern

that's a representative of the generator of $\pi_1(\Omega^1S^2)$? :P

Akiva Weinberger

03:59

Shame there doesn't seem to be any way to do it with other surfaces

Balarka Sen

@Akiva Yeah I think it has 6 hyperbolas

just smashed togather two index -1 things

Semiclassical

Which matches nicely with what I said earlier: pole + pole = dipole.

Akiva Weinberger

Actually, I think that picture can only make +2 poles, so it definitely can't work with other surfaces.

Balarka Sen

Which picture?

@arctictern Nice perspective

Semiclassical

Though with the subtlety that a 'positive' pole (CW rotation) and a 'negative' pole (CCW) both have the same index (-1).

Akiva Weinberger

04:01

The idea in the picture Semi just posted for finding the one-pole field on the sphere can't work for any other surfaces because the poles have to look different there

Balarka Sen

Ah, yeah, sure.

Semiclassical

but you can do similar 'smashing' in order to get stuff like quadrupole fields.

I should also point out that, in the picture I had earlier, the index would be -1/2

(I think)

Balarka Sen

Eh, index can't be non-integral

Semiclassical

I think the difference here is that you're not considering maps from the plane to S^1, but from the plane to RP^2.

Balarka Sen

Oh, that weird thing.

Mike Miller

04:03

what are we doing

Semiclassical

Yeah.

28 mins ago, by Semiclassical

Apropos of mostly nothing, this is a neat picture.

rambling based on that.

Akiva Weinberger

@MikeMiller We were discussing the one-pole vector field on the sphere (and Semi provided an illustration of the basic idea of the construction)

Semiclassical

and Balarka brought up vector fields on a sphere.

Balarka Sen

@MikeMiller Elaborate procrastination. Poincare-Hopf stuff.

Semiclassical

^

Amusingly, this stuff does show up in actual physics experiments (as the picture I linked is an example of)

Akiva Weinberger

04:05

And then I asked what the one-pole vector field on the double torus looks like.

Semiclassical

Another fun picture: goo.gl/images/LbNh2K

Balarka Sen

(That's the theorem which says index of any vector field is conserved as is equal to the Euler characteristic of the manifold, @Akiva)

Akiva Weinberger

Also, you know that cool picture that shows how there's a universal covering map from the hyperbolic plane to the double torus?

@BalarkaSen (Yes, I know.)

Balarka Sen

Yep

Akiva Weinberger

So what does the lift of that field look like on the hyperbolic plane?

Semiclassical

04:07

Alas, I don't remember what interpretation of that texture is. I know that the 'points' on there are local topological defects

Balarka Sen

It should have the pole situated on one vertex of each fundamental polygon

I think

Semiclassical

but I don't remember quite how that works.

Akiva Weinberger

This image. I forgot that the octagon was so large.

@BalarkaSen Could be anywhere in the polygon, I think

Semiclassical

oof

Akiva Weinberger

@Semiclassical So places in the crystal where there's an odd number of half-turns as you go around them are called "topological defects"?

Balarka Sen

04:08

Yeah, I am just trying to see if I can situate it near the vertex to have the six hyperbolas inside

Akiva Weinberger

Because you can't possibly get rid of them through continuous changes?

Semiclassical

Yeah.

some light discussion of topological defects here: lassp.cornell.edu/sethna/OrderParameters/…

Akiva Weinberger

The Klein–Beltrami model version of that image probably has an even larger octagon. (That's the one that has straight lines for hyperbolic lines, but destroys angles and circles)

Balarka Sen

Right. I dunno much about the geometry of $\Bbb H^2$; maybe I should

Semiclassical

my knowledge of hyperbolic space is limited to Escher pictures of the disc. (and some implicitly in special relativity, i suppose)

Balarka Sen

04:13

What I find nice about Poincare-Hopf is it's analogy with Gauss-Bonnet. Local things sum up to global thing which is a topological invariant of the manifold (in fact both turn out to be the Euler characteristic)

(G-B can actually be proved using P-H)

Akiva Weinberger

Here is the image I was looking for — a good illustration of the Beltrami–Klein model.

In the usual Poincaré model you're used to seeing, those triangles are a lot smaller.

Here, lines are preserved, but not angles or circles (they become ellipses).

Balarka Sen

Yes, I have seen that picture

Mike Miller

which is why they're both cases of atiyah singer

Kasmir Khaan

hey guys , can someone tell me the idea of the proof of the jacobian determinant in a variable substitution ?

Balarka Sen

@KasmirKhaan It's a measure of how much the volume of a small cube gets distorted by the map you're substituting with. Derivative measures precisely that: the amount of distortion that a certain map makes on an infinitesimal cube.

@MikeM I don't actually know the statement of A-S.

Semiclassical

04:21

I've seen various 'little' index theorems.

But I won't pretend I have any understanding of the full A-S theorem.

Kasmir Khaan

@BalarkaSen we are asked to prove this on the exam between 11 others theorems

where can I find a good proof of that?

thanks again balarka

the proof should explain the idea of the map of area from u-v to xy

Semiclassical

So you're just interested in the 2-variable case?

Kasmir Khaan

yes @Semiclassical

iv seen some proofs with partitioning the area into small rectangles

Semiclassical

Yeah.

Kasmir Khaan

but its a lot to write and its hard to understand imo

is that the only way?

Balarka Sen

04:24

The proof of change of variables theorem? Ted Shifrin sketches a proof in one of his lectures; I don't know an easily available source.

Kasmir Khaan

if so can you help me put the"pieces" together , i dont want to remember proofs i want to understand

we also have in our book such proof but they ask for more detailed one ><

Semiclassical

I'd say "look in any multivariable calc book" but the levels of detail vary.

Kasmir Khaan

yes i should =p

thanks again guys

can you give me the big picture of that partitioning propf?

i mean a way to understand it so i remember all the steps

Semiclassical

I'll be honest, I don't remember the proof anymore. It's one of those things that you learn to take for granted.

Balarka Sen

There is a very detailed proof in Shifrin, "Multivariable Mathematics" but it's not an easily available book

Kasmir Khaan

04:28

okay thanks guys :)

Semiclassical

This looks reasonably detailed: math.dartmouth.edu/archive/m13w12/public_html/notes/class12.pdf

Kasmir Khaan

Thanks ! ill read it now :D

Secret

I have finished re-reading the Wheels paper. I pretty much understood most of it (the universal problem is such a useful way to check whether an algebraic structure can be defined or not, and the logical structures of equations allow you to 'derive' the axioms and the details of the binary operators. I am also convinced that wheels extend semirings into involutive monoids, and thus semirings are embedded within wheels

However, those weird wheel axioms concerning zero terms still took some time to get used to. They seemed to be popping in and out randomly without pattern

Akiva Weinberger

@Secret I haven't looked at the paper in any detail, but one of their examples is solving an equation in Z/nZ where you don't know what n is, right?

So you have to add in zero terms every time you divide by anything, because it might be n.

Maybe the intuition comes from looking at that sort of thing

Secret

Yes. By bubbling in and out zero terms, and solving x in terms of some number + zero terms, once it is realised that the structure of interest allow cancellation of a given x, then the zero terms of the form 0/y disappears into the additive identity. In that work, however they make heavy use of those wheel axioms in sucession, I have recently wrote an MSE question trying to ask how to interpret them as all except one don't seemed to follow the usual distirbutive law

the introduction of zero terms and in general reciprocal elements allow one to work with modular systems without need to check whether it is cancellable, as the work will be taken care of by the disappearance of the zero terms

0

Q: Wheels. Nature of the zero terms?

In Wheels, zero terms were introduced by extending commutative semirings into an involutive monoid. The following rules/axioms were given in the definitions \begin{align} xz+yz & =(x+y)z+0z\\ (x+yz)/y & =x/y+z+0y\\ (x+0y)z & =xz+0y \\ /(x+0y) & = /x +0y \end{align} While the first rule is still...

Wheels is actually a very successful example of a zero term algebra IMO, they really only gave up 0/x=0 for all x and then get all sort of nice properties including structure preservation of semirings

This is the example that you referred to. It still look very alien to me due to the zero term axioms in wheels

Meanwhile, the meadows papers are still intractable to me. I am currently rereading a zero term algebra attempt written by the user Jonathen Cenders.

In addition,In the recent literature 2010 , there's a paper that basically claim 1/0=0. I have not read that one in detail yet however. That one is paywalled thus I need to use my uni library to read it

The following is a snipplet of Cender's pdf of his attempt

It is clear by reading many examples of zero term algebras, there is one thing in common: In order to make a divisible zero (division by zero meanwhile is still a topic of research), zero terms must be introduced. This is common in wheels, in absence numbers and in meadows

Balarka Sen

05:00

@Secret I really think the new "don't hog" guidelines here should be followed.

user228700

Hello, everyone :-) I have a quick question related to dimensional analysis (physics). For three given quantities having some powers of $M$, $L$ $T$, I must find if these three quantities are related by multiplication/division/power relation.

arctic tern

What?

user228700

Hang on, I am writing an example to illustrate the point.

arctic tern

@BalarkaSen the thread on main has been updated

in any case I put an exception there for when it's slow

(not sure if you and semi were done with your convo, wasn't following)

user228700

Let the three quantities be Pressure, $P$, Density $\rho$ and Velocity $v$. They are expressed as powers of $M$, $L$ and $T$ as follows:

Secret

05:09

I will use more [text]() in the future, and try to write more stuff in a few lines and less wordy

user228700

$P= M^1 L^{-1} T^{-2}$

Balarka Sen

I just think the message-salad shouldn't be so large as to cover the whole screen; eg links can be handled more carefully

user228700

$\rho = M^1 L^{-3} T^0$

user228700

$v = L^1 T^{-1}$

Mike Miller

@Balarka It gives topological data on one side which can often be represented as an integral over local data and an analytic side which is global.

Balarka Sen

05:11

Ah

Mike Miller

Lurie espouses the view that the existence of a local-to-global formula for the Euler characteristic is the same as a simple homotopy type.

user228700

Clearly, $P= \rho v^2 = (M^1 L^{-3} T^0){ (L^1T^{-1})}^2 = M^1 L^ {-1} T^{-2}$

Balarka Sen

(Which can also be guessed /cheating/ if you remember the Bernoulli principle of hydrostatics)

user228700

(Yes)

Balarka Sen

@MikeMiller Interesting, how so?

user228700

05:13

My goal is to find out if three (or more) such quantities expressed as powers of $M$, $L$ and $T$ are related (without knowing a relation beforehand, obviously)

Mike Miller

Giving a homotopy type a description as a CW complex (and thus a formula for the Euler characteristic) specifies a simple homotopy type. Similarly with a smooth structure on a manifold.

It's somewhat more sophisticated than that.

arctic tern

@Kaumudi.H if you want a systematic way of doing this, take logarithms of both sides and then solve the linear system

Balarka Sen

Hmm, that kind of makes sense

user228700

For three quantities, what my textbook has done is, it has expressed the powers of these three quantities as elements of a $3\times 3$ determinant.

user228700

> "This determinant evaluates to 0, hence these three quantities are not independent"

user228700

05:16

Why does this work?

user228700

@arctictern ...trial and error, right?

Balarka Sen

Linear systems are solvable iff the corresponding matrix is nonsingular

arctic tern

@BalarkaSen well...

user228700

But what linear system are we talking about, in this case? @Balarka

arctic tern

@Kaumudi.H did you read my comment about taking logarithms?

Balarka Sen

05:17

hm, @arctic?

user228700

@arctictern Yes, I did...unless I know what terms are on the R.H.S and what are on the L.H.S, how will this work?

Balarka Sen

taking logs is not quite necessary; just manipulate with the exponents.

If $A$, $B$ and $C$ are your quantities, write $A^a = B^b C^c$ and then write $A$, $B$ and $C$ in terms of $M$, $L$ and $T$.

arctic tern

@Kaumudi.H um, you do know both sides of the equations don't you? you even wrote out the equations for us.

Mike Miller

the thing I linked is short and readable.

arctic tern

@BalarkaSen your claim is wrong. systems with nonsingular matrices can have solutions.

claim needs amending

Balarka Sen

05:19

@arctic Oh, I meant uniquely.

user228700

@arctictern No, that was just an example. Usually, I dunno this.

arctic tern

@Kaumudi.H then I don't understand what the problem is.

You're not solving for $M,L,T$ in terms of $P,\rho,v$?

Balarka Sen

@MikeMiller Ok, looking

user228700

@arctictern No, not always. I will be given random variables with random powers in $M$, $L$ and $T$ and asked to check if they're related so I won't know what's on the L.H.S and what's on the R.H.S

arctic tern

@Kaumudi.H so you are given the equations.

Semiclassical

05:20

e.g. find a combination of P,rho,v which gives units of energy (M.L^2/T^2)

user228700

@arctictern What equations? I will only be given the individual variables. I will have to come up with an equation on my own... if that's even possible.

Balarka Sen

I explained how to get the equations above...

user228700

Yes, one second, @Balarka.

arctic tern

@Kaumudi.H "I will be given random variables with random powers in M,L,T" i.e. equations like $P=M^1L^{-1}T^{-2}$?

user228700

@BalarkaSen Right, so we have 3 equations and if they are solvable, then the quantities are related, yes?

user228700

05:23

@arctictern Well, no, I won't be given those equations either but I can easily deduce them. The problem is that I will have to relate the random variables using a single equation...well, check if such an equation exists.

arctic tern

@Kaumudi.H Okay, the problem is you have to relate the new variables (please don't call them random variables, unless you're actually referring to random variables). What information are you given about the new variables?

Semiclassical

If $P,\rho,v$ have the units you gave earlier, then $P^a \rho^b v^c$ has units of $M^{a+b}L^{-a-3b+c}T^{-2a-c}$

Agreed?

Balarka Sen

@Kaumudi.H Uniquely related, right.

user228700

@arctictern Sorry :-P I'm not given any information about the variables...well, they aren't variables, they're just symbols for physical quantities like pressure, density, velocity, etc but I dunno if they're related (unless there is some law that I have previously studied)

Semiclassical

You're given the units of those quantities, though.

Otherwise they'd just be meaningless symbols.

arctic tern

05:26

@Kaumudi.H Okay. I'm thinking of two variables, A and B. How are they related? I will give you no information about them whatsoever. I doubt this is the problem you have at hand.

user228700

@BalarkaSen So...they're solvable only if the determinant is not zero, no?

Balarka Sen

Uniquely solvable like arctic pointed out but yes

Semiclassical

If I tell you that I've got a viscosity specified in units of poiseuilles, that's not going to tell you much unless you either know that 1 poiseuille = 1 pascal*second or you know some physical law involving viscosity.

user228700

@arctictern No, not exactly, since I also know how these two are related in terms of some other variables such as $M$, $L$ and $T$ but if it's just two variables, the problem becomes ridiculously easy. I'm talking about three, maybe four variables.

Alias User

Would someone mind checking this proof briefly? mathb.in/118911

user228700

05:29

@BalarkaSen Huh. Why then has my textbook said "It evaluates to 0 so they are related"?

user228700

@Semiclassical Sure, yeah.

user228700

.__. What?

Balarka Sen

i'm confuzzled. that sounds wrong

Semiclassical

If someone tells me the speed of light, the distance from the sun to the earth, and the length of a year, then I have 3 dimensionful quantities with units of Length and Time.

Balarka Sen

i'll let @Semiclassical handle this though. he's the one who likes units

i chicken out

Semiclassical

05:32

Hah.

user228700

No, @Balarka, can u elaborate on just that one point?

user228700

Sigh. Okay, um, @Semi: All I wanna know is how the determinant method works.

Semiclassical

Alright. Let's go back to what I said earlier.

user228700

And what was that..?

Semiclassical

8 mins ago, by Semiclassical

If $P,\rho,v$ have the units you gave earlier, then $P^a \rho^b v^c$ has units of $M^{a+b}L^{-a-3b+c}T^{-2a-c}$

user228700

05:33

Right.

Semiclassical

So if I wanted to get a combination that had units of $M^\mu L^\lambda T^\tau$

then I'd have to choose a,b,c accordingly to get $\mu=a+b$ and so forth.

So the units + the exponents generate a linear system.

user228700

Oh, yeah! And in this case, $\mu$, $\lambda$ ad $\tau$ are all 0.

user228700

So it's a homogeneous system of equations.

Semiclassical

Well, if all three of those are zero, then you'd be able to form a dimensionless quantity from those three dimensionful quantities.

The question is, can we do that?

user228700

But that is what we're looking to do, no? If we're able to find an equation, take all the terms and put them on just one side and then the other side would be 1.

Semiclassical

05:37

Depends on what you're trying to do. If you wanted to find a specific combination of P,rho,v that gave units of, say, viscosity (pressure * time)

user228700

Oh, right. But no, that's not the case. For what I want to do, it's a homogeneous system. (If that's what I was doing, even viscosity would be one of the "variables")

Semiclassical

Sure. So, can we find $a,b,c$ such that $P^a \rho^b v^c$ is dimensionless?

user228700

Yeah, that is the question...the answer to which, um, hang on :-P

user228700

So, from what I've learned about homogeneous system of equations. if the matrix is...singular, only the trivial solution exists.

Semiclassical

Other way around.

If it's nonsingular, then the homogeneous system of equations is uniquely solvable. But the trivial solution always works, so if the matrix is nonsingular then that's the unique solution.

user228700

05:43

Oh, wtf.

user228700

Alright, @Semi: Thanks very much :-)

Semiclassical

It may help to think like this. Suppose I've got a diagonal matrix. In order for that to be nonsingular, every element on that diagonal must be nonvanishing.

user228700

Right..?

Semiclassical

Even one failure of that is enough to make it singular. So there are a lot of ways to make it singular.

user228700

Hmm, okay.

Semiclassical

05:46

And for each diagonal element that vanishes, I can take the corresponding variable (i.e. the kth diagonal element corresponds to the kth variable) to be nonzero

and it still would lead to a solution of the homogeneous system.

user228700

Oh, riight.

user228700

Alright, thanks very much, @Semi _/\ _

Semiclassical

Mmkay. So, is the matrix associated with P,rho,v singular or nonsingular?

user228700

It's singular. So yes, they're dependent :-)

Semiclassical

Yep. So we can form from these quantities the dimensionless ratio P/rho*v^2. In hydrostatics, that'll be exactly one.

But you could also write down that combination in, say, turbulent flow. It'd still be dimensionless, but it probably wouldn't be 1 anymore.

Sidenote: There are a lot of dimensionless numbers in fluid mechanics

That particular one looks to be the Euler number.

Wikipedia also has a similar combination under the name of pressure coefficient.

user228700

05:53

Oh, right. Fluids.

Semiclassical

Yeah.

One useful exercise might be to take one of those numbers on there and convince yourself, via the determinant method, that their physical parameters indeed generate a singular matrix.

user228700

Yes, I will do that.

user228700

Again, thanks very much :-)

Semiclassical

Another thing to note is that the determinant method is binary: it doesn't say 'how badly' a matrix fails to be nonsingular, it just says singular or not.

As an example of where that could become a problem, suppose I included viscosity $\mu$ with its units of pascal*seconds.

I could write down the 4-by-4 matrix, and it would again vanish. No surprise there.

What's worth noting, though, is that in that case there will be two independent dimensionless ratios, not just one.

The fact that the determinant vanishes tells you that there's at least one such ratio. But in order to know how many, you'll have to do something more elaborate with the matrix e.g. reduction to row echelon form.

user228700

Ohh, okay. Nah, that's okay, all I need to know is if they're related or not.

Semiclassical

05:58

Alright.

In applications, you typically do want to know 'how many' such independent relations there are. The reason is that the physical laws governing a system of physical quantities should only depend on dimension ratios.

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