Mathematics - 2019-01-22 (page 1 of 2)

12:00 AM

Guys, I have a question

when parameterizing a function, we create a mapping and then to eliminate the variable (we compose functions), how can I prove that the composition always yields the function?

Akiva Weinberger

Hm

vimeo.com/20287194

How can we describe these curves mathematically

I'm guessing the computer is solving a differential equation of some sort

ÍgjøgnumMeg

Right, I did a course on PDEs in my undergrad and we plotted the solution to a Bessel equation. The solution looked e x a c t l y how you'd expect a swinging chain to look, which was magical to me

Akiva Weinberger

Or at least, it's modeling the wire as a bunch of line segments, and it's computing some value in terms of the angles between the line segments ("bendiness?") that it's trying to minimize

@ÍgjøgnumMeg Cool

I know nothing about the Bessel equation

though I guess it shows up in physics, the way you're describing it

ÍgjøgnumMeg

yeah that's the only context I've seen it in, as a solution to a particular form of ODE

as in, Bessel functions

Akiva Weinberger

The bent-wire curves almost look like Bézier curves

but they're probably more complicated than that

Leaky Nun

12:06 AM

@AkivaWeinberger reading about curves recently?

Akiva Weinberger

(Bézier curves, by the way, are made from repeated linear interpolation, and they're a really neat idea)

@LeakyNun Well it's a long rabbit hole that started with minimal surfaces

("Soap film surfaces")

Leaky Nun

have you been doing any "proper" maths?

mathsssislife

Guys i'm not considering cases where x=t and y=f(t) where the original equation is y=f(x)

i'm considering equations where this is not the case

LWChris

For a moment I though all the quotes in this chat were from a guy named "jan"... I should quit working for today

Akiva Weinberger

@mathsssislife I'm not sure I understand what you're composing

@LeakyNun There's this really neat minimal surface that repeats in a cubic lattice, called a "gyroid", and it looks really weird

It's used as an infill in 3D printing

(It performs similarly to some other infills, in terms of printing time, weight, and strength, but it beats them all in how cool it looks)

^Statues of the gyroid (which you can buy from Bathsheba Grossman)

ÍgjøgnumMeg

12:10 AM

@Akiva one of my lecturers took a lot of iterations of a construction for a spacefilling curve and them superimposed them on top of each other, connected it all up and 3D printed it

which looks cool

mathsssislife

@AkivaWeinberger Consider the function y=f(x): i'm considering cases for instance the function, x=g(t) and y=h(t) how can I prove h(g(t))=f(x)?

Akiva Weinberger

Animated gyroid structure

►

mathsssislife

never mind

Akiva Weinberger

(When you see a time lapse of it being 3D printing, it looks like that^ but in reverse)

The gyroid partitions space into two pieces. The pieces are called "labyrinths"

Here's a "skeleton" of the labyrinths

vimeo.com/2553639

(so the gyroid weaves between the "skeletons")

Not sure what the green things at the end are. Paths that avoid the gyroid?

I think they go down those spiral paths you see in the skeletons

The spirals are why it's called a "gyroid", by the way

'Cause they "gyrate", I guess? Not 100% clear on that

Nah never mind, the green lines go through the vertices of the skeletons. So I'm not quite sure what they are

Oh, he describes it in the description

"Triply periodic minimal surfaces" is the phrase to Google, by the way

since it repeats in the x, y, and z directions

Triply periodic minimal surface

In differential geometry, a triply periodic minimal surface (TPMS) is a minimal surface in ℝ3 that is invariant under a rank-3 lattice of translations. These surfaces have the symmetries of a crystallographic group. Numerous examples are known with cubic, tetragonal, rhombohedral, and orthorhombic symmetries. Monoclinic and triclinic examples are certain to exist, but have proven hard to parametrise.TPMS are of relevance in natural science. TPMS have been observed as biological membranes, as block copolymers, equipotential surfaces in crystals etc. They have also been of interest in architecture...

People are trying to classify them all but I think they haven't yet succeeded

mathsssislife

Guys what is the formal definition of a parametric equation?

Mike Miller

12:22 AM

what's the informal definition of a parametric equation?

mathsssislife

or a parameter

Rithaniel

I kind of wanted to get the gyroid sculpture from Bathsheba, but the 4" version, is, like, $400.

Is there a term for a space which is "gyroid-like" in geometry, but instead of 2D surfaces bending through 3D space, it's 3D through 4D space?

(Perhaps a bit too inexact of a description)

Akiva Weinberger

If y is some function of x and you represent it parametrically by (h(t), g(t)), then g(t) is that function of h(t) @mathsssislife

Think of it in terms of graphs - for all values of t, you're drawing the point (h(t), g(t)) onto the graph

Compare that to y=f(x), where for all x you draw (x, f(x)) onto the graph

@Rithaniel Quadruply periodic minimal 3-manifolds?

Instead of triply periodic minimal surfaces

Rithaniel

Ah, yes, perfect.

And more specifically, is there an example of such a manifold or family of such manifolds which have similar symmetries to a gyroid?

Akiva Weinberger

@Rithaniel Luckily, the pictures are free

Rithaniel

12:33 AM

Indeed they are.

Akiva Weinberger

@Rithaniel This stuff is hard as it is in 3D - you want this in 4D?!

I mean, if some crazy mathematician has attempted this then I wouldn't be too surprised

or maybe it's easier than I think, who knows

Rithaniel

That might be interesting to conduct some research into that. Perhaps a generalized form of $n$-periodic minimal $m$-manifolds for $m<n$

mathsssislife

How do I prove that eliminating the free variable t in a parametric equation yields a unique function?

Akiva Weinberger

I need to go to bed

Before I do, let me just point out that parametric things don't actually need to satisfy the vertical line test like functions do

mathsssislife

yeah I know, but i'm wondering as to why if they do satisfy the vertical line test then eliminating the variable yields y=f(x)

the function which they trace

i mean

Sir Cumference

1:51 AM

Dumb question, but can every nonempty finite set be well ordered?

Sir Cumference

2:09 AM

@Dair Nice name lol

Dair

2:25 AM

@SirCumference Nice name lol

Daminark

So, every set can be well-ordered in ZFC. If you're using another system maybe something weird can happen. Though regardless of one's opinions on ZFC, finite sets have an honest-to-God well-ordering since they biject to $\{1,\ldots,n\}$, which has its natural well-ordering

Sir Cumference

@Daminark Huh seriously? I may be misunderstanding, but isn't a well order a total order such that every subset contains a minimal element?

How would you define a well order on $\mathbb{R}$?

Wait sorry, for clarification did you mean "every finite set"?

Daminark

Nah, every set, this is called the well-ordering theorem, it's equivalent to the axiom of choice

Do you know Zorn's lemma?

Sir Cumference

Nope, I've only touched the surface of order theory :/

The definition of a poset, total order and well order are about what I know

Daminark

So, Zorn's lemma is the statement that if you have a poset such that every totally ordered subset is bounded above, then there's a maximal element

Sir Cumference

2:39 AM

@Daminark A maximal element of each totally ordered subset?

Daminark

But yeah, so the axiom of choice, Zorn's lemma, and the theorem that every set can be well-ordered are all equivalent

Maximal, it just means there's an element of the set such that no element is strictly larger, not necessarily that it's larger than everyone else

Sir Cumference

I mean we're talking about the subsets having maximal elements right?

Daminark

No, the poset has a maximal element

Sir Cumference

@Daminark Interesting. Do you recommend an introductory textbook on these topics? My uni doesn't have an in-depth course on this stuff

Daminark

I don't really study this stuff myself much, I'm vaguely aware of the absolute basics of set theory and that's about it

Halmos' Naive Set Theory is a book I've heard of

Also Jech has a book that's very hard

Sir Cumference

2:43 AM

@Daminark I maybe being pedantic, but if an element is not comparable to anything else, then nothing is strictly larger than it. Do we consider that to be maximal?

Daminark

I mean yeah that works

Sir Cumference

@Daminark I see, thanks

Daminark

No problem

But yeah so, those books I mentioned are if you specifically care about set theory

There's an amount that every mathematician should know but that amount is very tiny, and if you're only just looking for that amount you can prob just read chapter 1 of some analysis or algebra book floating around

Sir Cumference

I mean yep, set theory and logic and other foundational topics are interesting to me. Shame they are overlooked by our math dept.

@Daminark Yeah, I'm mostly around that minimal amount (at least I think).

Silent

We get done with just 3 matrices, because, in our examples, one entry is 1. But, generally, $\begin{bmatrix}
1 & 0 \\
0 & c \\
\end{bmatrix}
\begin{bmatrix}
b & 0 \\
0 & 1 \\
\end{bmatrix}
\begin{bmatrix}
1 & 0 \\
\frac{d}{c} & 1 \\
\end{bmatrix}
\begin{bmatrix}
0 & 1 \\
1 & 0 \\
\end{bmatrix}
=
\begin{bmatrix}
0 & b \\
c & d \\
\end{bmatrix}$

Dair

3:06 AM

@Daminark Clearly, it's all about type theory

type theory is what the real applied metamathematicians use.

but how many schools teach type theory?

Le Anh Dung

4:35 AM

Hi everyone! I have asked a question about a closed formula of Cantor set here. It seems to me that nobody is interested in this question. Could someone help me resolve the question? Thank you so much!

Dair

4:46 AM

@LeAnhDung I feel like i've done this problem a while ago. Usually when showing two sets are equal I will also some time resort to the following tactic: Let $A_n = \bigcap_{m=0}^{n}\bigcup_{k=0}^{\lfloor 3^m/2\rfloor}\left[\frac{2k}{3^m},\frac{2k+1}{3^m}\right]$... Suppose $x \in A_n$? Is $x \in C_n$? Then consider the other way round.

Tanner Swett

5:06 AM

Yo. I've noticed that certain questions about mathematics are "ontological questions", in the sense that the answer depends on the ontology you've chosen for mathematics.

An example of an "ontological question" is: how many trivial groups are there?

In ZFC, there are so many trivial groups that the class of all trivial groups is a proper class.

In homotopy type theory, there is exactly one trivial group.

Another similar one is: let $G$ be a group isomorphic to the Klein four-group. How many groups are subgroups of $G$ of order 2?

In ZFC, the answer is 3; in homotopy type theory, the answer is 1.

The reason for all this is that in ZFC, given any algebraic structure (which contains at least one element), it is possible to find a proper class of distinct algebraic structures isomorphic to it; whereas in homotopy type theory, there are never two algebraic structures which are distinct but isomorphic.

Al t.

Hi. Could someone help with this please? math.stackexchange.com/questions/3082751/…

Tanner Swett

Anyway, I'm wondering if there are any questions and answers posted about this general topic.

Al t.

Sorry Tanner, I interrupted your question.

Tanner Swett

Not a problem. :)

Karl Kronenfeld

@Tanner Don't know about questions on this site, but such ontological questions are considerations that motivate structuralism in the philosophy of mathematics. See e.g. plato.stanford.edu/entries/philosophy-mathematics/#StrNom

Mike Miller

5:30 AM

@TannerSwett There is 0% chance this is correct, and must amount to a misunderstanding of the notions in homotopy type theory. It will never be the case that a different foundational setting should change fundamental statements about a theory like this that aren't just artifices of those foundations (like the proper class vs one above).

What you are probably looking for in the homotopy type theory setting is something like the following. A subgroup of a group $G$ is defined to be a group with an injection into $G$ (or something like this). An isomorphism between such exhibits them to be the same.

But an isomorphism just of the subgroup is not sufficient.

The other stuff (proper classes vs one particular thing) is legitimate and worth asking about.

Tanner Swett

@MikeMiller I intentionally wrote "how many groups are subgroups of $G$" instead of "how many subgroups does $G$ have"; did you notice that?

Mike Miller

eh, I see your point.

👍

Tanner Swett

In any case, you've hit on the exact distinction I'm talking about: the distinction between statements which are meaningful independent of foundations, and those which are "just artifacts of those foundations".

Mike Miller

I'm trying to figure out how to phrase the question you want to ask

it's a good one, and one a friend has talked to me about before

something along the lines of "how do I avoid asking ontological questions?" seems interesting to me, or possibly "how do I distinguish" them

Tanner Swett

5:46 AM

I think I want to post it as a self-answered question. Because I think I know the answer; I just have to figure out what the question is. :D

The answer is something like:

"The answer depends on the foundation you're using for mathematics. Different foundations will give you different answers. None of these answers are wrong; this is simply a question whose answer differs from foundation to foundation."

Mike Miller

I don't think that's the answer to the right question.

The problem, I think, is that ontological questions (by their nature) are irritating little creatures which don't actually relate to the theory you're trying to talk about. So instead of worry about whether any answer is right or wrong, it's more interesting to talk about how to pin down those questions that avoid this problem.

I guess that's just to my eye.

I think of this as being quite distinct from situations like the Whitehead problem in which the answer is independent of standard foundations, as opposed to being related being able to ask questions like "is $0 \in 2$?"

Tanner Swett

My point in bringing this up is that someone posted a question along the lines of: "Is every cyclic group a subgroup of some other group?"

Karl Kronenfeld

lol do it

Tanner Swett

And I thought: well, gee, given a group, it's not necessarily meaningful to ask whether or not it's a subgroup of some other group.

So ideally I would have posted a comment (or an answer) saying: hey, this question sounds like an "ontological question", and go look at this other question to see what I mean by that and why that's significant.

The problem is that I don't yet have an "other question" to link to.

Karl Kronenfeld

no just leave it up in the air

user76284

5:58 AM

@TannerSwett What's the difference between "how many groups are subgroups of G" and "how many subgroups does G have"?

Mike Miller

@user76284 He means to ask "how many groups are isomorphic to subgroups of G", essentially. In standard set-theoretic terms a group that is a subgroup of G is what you think it is, and he's not quotienting by isomorphism in the statement of the question.

Whereas the claim (which I can't say anything about) is that in HoTT the meaning of what it is to be a group is different (you don't start your life as a set), and so the answer to "how many groups are isomorphic to subgroups of G" should be the same as "what is the number of isomorphism classes of subgroups of G".

Tanner Swett

@user76284 Well, the Klein four-group has three distinct subgroups of order 2, right?

Correct me if I'm wrong, but in ZFC, those groups are distinct and unequal groups, because they all have different elements.

In homotopy type theory, there's only one group of order 2, so those can't possibly be 3 different groups.

Actually, let me explicitly define "subgroup" and "group which is a subgroup".

For ZFC: If $G$ is a group, then a group $H$ is a subgroup of $G$ if the underlying set of $H$ is a subset of the underlying set of $G$, and the operation on $H$ is the appropriate domain restriction of the operation on $G$.

The definition of "group which is a subgroup" is the same.

For HoTT: If $G$ is a group, then a subgroup of $G$ is a group $H$, equipped with a homomorphism $H \to G$ which is an injection.

If $G$ is a group, then a "group which is a subgroup" of $G$ is a group $H$ such that there exists a homomorphism $H \to G$ which is an injection.

I'm not even sure that my definition for ZFC is correct; I might have botched it.

The weird thing about homotopy type theory is that generally, algebraic structures are defined in such a way that if a structure $G$ is isomorphic to a structure $H$, then $G = H$. So it's not necessary to say "up to isomorphism", because everything is up to isomorphism; and there's no need for a concept of "isomorphism classes", because any isomorphism class would only contain one element.

And yes, in homotopy type theory, there is only one set of each cardinality.

Akiva Weinberger

6:36 AM

vimeo.com/11993047

3

@Silent Oh yeah, the (1 2 \\ 3 4) one had a 1 that was used as a shortcut

Akiva Weinberger

6:53 AM

Trefoil medial surface

(Locus of points with at least two nearest neighbors on the trefoil)

For a long time, I was trying to think of a good term that describes a rectangle whose sides are parallel to the axes. I think I finally found a good one: "axis-aligned"

Leaky Nun

8:24 AM

@TannerSwett In HoTT, If you look at injections $H \to G$, instead of just $H$, then I believe you get 3 subgroups of the Klein group

Akiva Weinberger

8:39 AM

This sounds like category theory

@LeakyNun They're equivalent modulo (outer) automorphism, dunno if that's relevant

You're looking at $\Bbb Z_2\hookrightarrow V_4$?

Leaky Nun

yes

Akiva Weinberger

I like how, if you view it as the symmetries of a rectangle, you get it as a subgroup of $S_4$ in two ways

As a permutation of the edges, it's $\langle(12),(34)\rangle$ (aka $\Bbb Z_2\times\Bbb Z_2$)

As a permutation of the vertices, it's $\langle(12)(34),(13)(24)\rangle$ (which is a normal subgroup of $S_4$)

Leaky Nun

@AkivaWeinberger nice

Akiva Weinberger

8:55 AM

Hm, it's also a subset of the symmetries of the cube, yeah? The permutations that preserve the pairs of opposite faces

aka the 180 degree rotations (plus identity)

And that makes the outer automorphisms more obvious

Same thing for the symmetries of the tetrahedron

@AkivaWeinberger About the square faces

Leaky Nun

@AkivaWeinberger the symmetries of the cube is just S4 so sure

Akiva Weinberger

Oh right so then we have the two ways of getting the Klein group out of the cube then

One is 180 degrees about the faces, the other is 180 degrees about a face and the two perpendicular pairs of edges

Akiva Weinberger

9:20 AM

@LeakyNun Here's a puzzle worth thinking about

Say you have a function from the plane to itself that preserves distances of length 1.

That is, a function $f:\Bbb R^2\to\Bbb R^2$ such that, if $|x-y|=1$, then $|f(x)-f(y)|=1$.

Prove that it's an isometry (i.e. either a rotation, reflection, translation, or glide reflection).

(You may not assume that $f$ is injective or continuous)

Leaky Nun

assume that $f$ is injective and continuous. then it follows :P

Akiva Weinberger

Does it?

It's not obvious even with that. Why are distances of length 1/2 preserved?

Or length 2?

Leaky Nun

I'm just joking

Akiva Weinberger

This also works in 3D and higher, with the same conclusion, by the way

but my solution for 3D is slightly different than my solution for 2D

Alessandro Codenotti

10:50 AM

@LeakyNun looks like I might learn some lean next semester after all

Akiva Weinberger

Boing

Leaky Nun

11:06 AM

@AlessandroCodenotti wth :o

welcome to the club!

you might want to register on the lean chatroom

Akiva Weinberger

Alessandro Codenotti

Well I'm still not 100% sure about which courses I'll take next semester, but type theory sounds extremely likely

Akiva Weinberger

Ugh

Leaky Nun

@AlessandroCodenotti I would be extremely happy to see you in the club!

Alessandro Codenotti

Where is that chatroom you mentioned?

Leaky Nun

11:15 AM

@AlessandroCodenotti leanprover.zulipchat.com

Akiva Weinberger

There we go

It's called "Vogel's floret". Basically, you take Fermat's spiral ($r=\sqrt\theta$) and periodically place a circle

at the "golden angle"

$360^\circ/\phi^2\approx137^\circ$

What it looks like with other angles (on the left and right)

Hm, question

Say we have a cuboid (i.e. the same combinatorial topology as the cube, but it can have irregular faces)

(Basically eight points such that certain groups of four are coplanar)

If seven of them lie on a sphere, does the eighth one?

Akiva Weinberger

11:44 AM

^Polyhedra made of rectangles

An orthogonal polyhedron is a polyhedron whose faces all meet at 90 degree angles. These^ are not orthogonal, despite having rectangular faces.

This is impossible for genus 0 and 1. Not sure about anything between 2 and 6 - the polyhedra in the picture have genus 7.

eurocoder

12:04 PM

I have the following summation: \sum_{j=1}^\infty 4/\alpha_j^4 where alpha_j is the jth zero of the Bessel function of order 0. This sum converges extremely quickly to 0.125. So does anyone know how this happens, are there known formulas for the summation of zeros of Bessel functions?

Perturbative

1:43 PM

Quick question, what does it mean for a group to be torsion? Like if an author says "$G$ is torsion" what does that mean?

Alessandro Codenotti

Every element has finite order

Perturbative

Ahh okay

Learner

1:58 PM

Hey everyone, does somebody know how we can exclude $\dim(\ker f^2)=\dim(\ker f)$ in the following?

0

A: Finding ch. polynomial and Jordan normal form of $f$ knowing $\dim\ker f=2$ and there are $a,b$ not in $\ker f$ such that $f^2(a)=0, f(b)=b$

Your answer is correct and your reasoning is mostly correct. However I don't understand how you can conclude that $f(a), b$ form a basis for a $2 \times 2$ Jordan block? For starters they have different eigenvalues so they can't possibly be in the same Jordan block. Want you want to say is the fo...

Jake Rose

2:20 PM

For stoum Louisville form do you need the function inside the derivative to be >?

Shaun

2:35 PM

My very own copy of Dummit & Foote has just arrived! $\ddot\smile$

Zee

5:17 PM

That’s a terrible book

3

ninja hatori

5:53 PM

Anyone familiar with hermitian forms

Astyx

Ask your question, we'll see

ninja hatori

I understand the defination of involution and its examples but I don't get that paragraph after examples are done. Why identity is involution ? and how involution change according to commutative property

@as

@Astyx

Astyx

The identity is not necessarily an involution, as stated

In fact it is only when the ring is commutative

ninja hatori

why it is involution in commutative ?

Astyx

Because for it to be an involution we want ab = id(ab) = id(b)id(a) = ba

for all a and b

ninja hatori

6:03 PM

Also why fixed elements form subring?

for commutative rings?

Astyx

Because if a*=a and b*=b, then (a+b)*=a*+b*=a+b

and (ab)* = b* a* = ba=ab

ninja hatori

and what is inverse by the way?

additive inverse

-

@Astyx thanks

np

6:08 PM

@Astyx what are you studying by the way?

Astyx

engineer degree

maths, computer science and physics

ninja hatori

ok ; familiar with pure mathematics or not?

Astyx

a bit

Ted Shifrin

salut @Astyx

Astyx

Salut

Quoi de neuf ?

Ted Shifrin

6:11 PM

Pas grand'chose, et toi?

Astyx

Pareil

Ted Shifrin

toujours en bonne santé?

Astyx

Oui ça va

Il commence à faire froid ici

On a eu les premiers flocons

Ted Shifrin

oooh, bien épatant!

Astyx

Et toi ?

Ted Shifrin

6:20 PM

J'ai mal au cou, mais autrement ça va, oui. Quelles maths apprends-tu ces jours-ci?

Astyx

Le cours est intitulé : "Distributions, analyse de Fourier, équations aux dérivées partielles"

Ted Shifrin

Ah, un cours bien intéressant!

Astyx

Et j'ai aussi un cours de maths appliqués "Modélisations de phénomène aléatoires, : introduction aux chaines de Markov et aux martingales"

Ted Shifrin

J'en connais très peu.

ninja hatori

How one can prove bijective isometry of hermitian module forms unitary group ?

Ted Shifrin

6:31 PM

Sounds like definition to me.

You mean hermitian vector space.

ninja hatori

yes @TedShifrin

is it simply defination or require some argument?

@TedShifrin

Ted Shifrin

I don't know your definitions, so I cannot answer.

Of course, you need to know that the isometry sends $0$ to $0$, or it's false.

Or is your course assuming an isometry is linear to start with?

ninja hatori

Yes it is linear isometry some sort of generalization of bilinear form

we needed

Ted Shifrin

I do not understand.

Zee

Ted , can you tell me what to take next semester?

Ted Shifrin

6:38 PM

No.

Zee

Ted , can you not tell me what to take next semester?

ninja hatori

I already study symmetric bilinear form so I am generalize this things to hermitian case ; and linear isometry is assume in case of bilinear form so I guess in this case also

Ted Shifrin

@ninja: Start by proving that a unitary matrix gives you an isometry, and then see if you can go backwards.

Zee

How important is knowledge of the following for you , Lie groups, SCVs , Rational homotopy theory

Ted Shifrin

@Zee: My interests are not relevant to you.

Talk to your adviser and other faculty there.

Zee

6:41 PM

Well , I wanted your opinion since your always mean to me. So I was gonna do the opposite of what you advise

“Don’t jump off a bridge then” HA

ninja hatori

@Zee not the right way to talk

Manolis Lyviakis

7:11 PM

anyone can help me with some differential geometry to understant something?

here is my confusion

consider 2 surfaces and a function between them. Now in order to define its derivative i will do it through the surfaces patches . And the derivative will be a linear transformation from the tangent plane of the first surface to the tangent plane again . And its matrix will be the matrix which passes through the basis defined by the partial derivatives of the first patch to he second

so if ψ,φ the 2 different patches of the 2 surfaces one basis will be ${ψ_u,ψ_v}$ and ${φ_u,φ_v}$ and the matrix will be $D(ψ^{-1} \circ φ)$ (jacobian of the composition ) acting on these basis

is that it?

i mean if i want to calcuate that matrix i calclulate the jacobian of the composition of the given patches then act that jacobian matrix to one basis find a vector write it as a linear compination of the other basis and put is coeficients as a collumn and the resulting matrix is the matrix of the derivative of the function going between the 2 given surfaces?

Érico Melo Silva

$\psi^{-1} \circ \varphi$ doesn't make sense, and what you wrote doesnt use the function between the two surfaces so it has to be wrong

Manolis Lyviakis

yes you are right

Érico Melo Silva

Let $f: S_{1} \to S_{2}$ be the desired function between the two surfaces $S_{1}$ and $S_{2}$, and $\varphi : U \subset \mathbb{R}^{3} \to S_{1}$, $\psi: V \subset \mathbb{R}^{3} \to S_{2}$ the local parametrizations

Manolis Lyviakis

$ D(ψ^{-1} \circ f \circ φ )$

Érico Melo Silva

yeah

that's the jacobian of $f$ written with respect to the coordinates induced by the local parametrizations

Manolis Lyviakis

7:26 PM

so i dont need to do $D(ψ^{-1} \circ f \circ φ ) *φ_u=a_11ψ_u +a_12 ψ_v$

just the jacobian is enough to say its the matrix corresponding to the derivative of f?

Érico Melo Silva

that doesnt even make sense bc $\varphi_{u}, \varphi_{v}$ and $\psi_{u}, \psi_{v}$ are bases of different vector spaces

Manolis Lyviakis

i was thinking i have to find $a_{11}, a_{12}, a_{21} a_{22}) $

Érico Melo Silva

Namely the first is the basis of the tangent space of $S_{1}$ at a point, and the second is the basis of the tangent space of $S_{2}$ at the image of that point under $f$

Manolis Lyviakis

since both are surfaces the tangent space is actually the same of rank 2

im not considering the general case between 2 manbifolds

Érico Melo Silva

yes, but what you're doing by conjugating the local parametrizations is your using the isomorphisms they induce on the tangent space at a point on the surface, to the tangent plane at a point in $\mathbb{R}^{2}$, and then using the fact that all the tangent spaces to $\mathbb{R}^{2}$ are canonically isomorphic to $\mathbb{R}^{2}$ to write down the matrix

Manolis Lyviakis

7:34 PM

so suppose i want to calculate the matrix correlated to the derivative of a function between 2 surfaces what do i do?

(essentially to caulculate the derivative)

Érico Melo Silva

you compute what $\psi^{-1} \circ f \circ \varphi$ is (that's just a map from a subset of $\mathbb{R}^{2}$ to a subset of $\mathbb{R}^{2}$)

Manolis Lyviakis

ok i do that

and then just the jacobian?

Érico Melo Silva

and then you just find it's jacobian

yeah you can do it cause it's just a regular map between open subsets of euclidean space

Manolis Lyviakis

hm.. why i i think thats wrong

Érico Melo Silva

work out an example and try what i said then

Manolis Lyviakis

7:37 PM

in order to calculate the matrix of a given linear transformation you must "act" that linear transformation the basis of your space

but the composition is not my linear transformation

if i wanted to find the matrix of T:W -> V i find the coeficients of T(e_1) , T(e_2)..

Érico Melo Silva

picking a basis of a real vector space is the same as giving an isomorphism to $\mathbb{R}^{n}$

that's what conjugating w the parametrizations does

Manolis Lyviakis

hm.. i dont quite get it but i got my answer

thanks alot

its weird thats its just the jacobian.

Mike Miller

It shouldn't be, that's the point of being able to work in charts!

Manolis Lyviakis

im always thinking that the jacobian uses the standard basis

ok then its pretty simple actually

Mike Miller

Don't think so much about bases, they don't matter.

Manolis Lyviakis

7:45 PM

although i searched alot calculating the derivative of a function between 2 surfaces

Mike Miller

There's a transformation law about what happens to the Jacobian if you change bases, or even if you change your chart by a diffeomorphism.

Manolis Lyviakis

didnt find a concrete example

Érico Melo Silva

maybe check Ted's book on diff geo

alpha.math.uga.edu/~shifrin/ShifrinDiffGeo.pdf

Mike Miller

I have to imagine you searched in the wrong places, then. :)

Érico Melo Silva

ch 2 probably has problems to this effect

Manolis Lyviakis

7:46 PM

who is Ted

Érico Melo Silva

he's the big kahuna

Mike Miller

just some differential geometry guy

Manolis Lyviakis

btw just looking from the at the contents thats exactly what i have to study

ill look into it

Mike Miller

Great!

Érico Melo Silva

(y)

oh wait that doesnt work here lol

Mike Miller

7:49 PM

i do it anyway

i copied the thumb on the starboard from google

Manolis Lyviakis

wait

is he avoiding talking about derivatives between surfaces?

Mike Miller

without looking: no

Érico Melo Silva

idr, if it doesnt have enough of what you need just check do Carmo's book or something

it's also on the internet

Manolis Lyviakis

its good really good

its exactly the things i need

Mike Miller

just remember that talking about the derivative of a map between surfaces/manifolds is just the same as talking about the derivative of a map between euclidean spaces, by looking at charts

Manolis Lyviakis

8:01 PM

ye i got it

he uses the notion of directional

derivative

for the shape operator

wish i had this pdf 3 months ago

so for example the unit normal vector is a function between 2 surfaces . Say for example a cylinder to the unit sphere

Érico Melo Silva

yes the gauss map

Manolis Lyviakis

now considering 2 patches and finding the jacobian must result to the matrix of derivative of the unit normal vector