Mathematics - 2018-07-18

Adam

00:20

Can anyone point me to the paper that Andrew Granville generalized Lucas's Theorem to prime powers and if it's paywalled send me the pdf in a pm?

Secret

00:33

[Random]

dx is an infintesimal

$dx^{j}$ is a tangent vector

$dx^{j}$ can capture an infinitesimal change in a function value, thus bypasses the non uniqueness problem of infintesimals as traditionally outlined by Lebniz

Now applying explosive generalisation, we can consider the following mathematical object:

$M(dx)$ is a mathematical object with the property being infinitesimal or capture an infinitesimal property, meaning:

$M(dx) = \{x \in M | \phi(x)\}$

where $\phi$ reads "is related to or has the property of being infintesimal"

Now recall the definition of an infinitesimal as an element $\epsilon$ in a nonarchimedian ring $R$ such that for any $x \in R$, $x\epsilon < 1$

hmm... need to figure out how to formalise $M(dx)$... perhaps the literature already have some examples

ÍgjøgnumMeg

10:05

@Adam web.archive.org/web/20170202003812/http://www.dms.umontreal.ca/… google is your friend

Manolis Lyviakis

11:28

hey

im studying dfferential geometry

and we construct and study the tangent space of a surface

but i dont see the reaso why we do that

Tangent space

In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector that gives the displacement of the one point from the other. == Informal description == In differential geometry, one can attach to every point x {\displaystyle x} of a differentiable manifold a tangent space—a real vector space that intuitively contains the possible directions in which one can tangentially pass through ...

how does that help us study the manifold itself?

the generalization of vectors from affine spaces to general manifolds what does that mean?

"the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds"

Daminark

So, a priori a manifold need not live in $\mathbb{R}^n$. It might just be some topological space kinda floating around and minding its own business

Manolis Lyviakis

ok

Daminark

The point is, you're trying to learn about manifolds via calculus

Manolis Lyviakis

the tangent space is usefull to define

Daminark

But calculus only makes any sense when there's a vector space structure to work with, so the tangent space is there to make sense of that

Manolis Lyviakis

11:34

the fundamental forms

ok

hm..

is it like studying

the derivative of the surface?

how that plane behaves from point to point it tells you how much the sphere curves? and so on.

Daminark

There's a sense in which that's the right way to think about it. For example, let's say you have some manifold which is just the level set of a smooth function. Then the tangent space of the manifold is just the kernel of the derivative

The way I tend to think about it is that if you have a smooth function between manifolds, then the derivative of that map at that point is going to be a linear map on tangent spaces

Manolis Lyviakis

yes

ohh

on tangent spaces

ohh nice didnt thought of that

Daminark

As for what you said about the tangent space telling you about curvature, that's something I know little about, I know some general facts here and there about manifolds and some topology but geometry is mostly outside of what I've been thinking about

Manolis Lyviakis

oh

Daminark

But yeah that sorta captures the idea of how this tangent space structure lets you do calculus, the derivative literally lives in these spaces. If you know anything about differential forms, say in $\mathbb{R}^n$, then they are defined out of tangent spaces, and that's the mechanism by which you want to talk about integration on manifolds

Manolis Lyviakis

11:42

oh ok . it helped me thanks

so they come up naturally

from the derivatives

(the tangent spaces)

Daminark

Yup, and no problem! :)

Manolis Lyviakis

link to see that

derivative of a manifold?(should i google that?)

derivative of multyvariable function?

Daminark

I think any book on manifolds will probably include that soon enough

You could Google something to the effect of "tangent spaces and smooth functions" to see what you get?

Manolis Lyviakis

ohh so i have the total derivative

of a function

$R^n$

now for a function

between manifolds

the derivative is called the differential

$df:Tdf(p)\colon T_{p}M\to T_{f(p)}N.$

$df(p)\colon T_{p}M\to T_{f(p)}N.$

which is a generalization of the total derivative

im startng to get it

Daminark

Fantastic!

Well, anyway, I think I'm gonna have to go now

Alessandro Codenotti

11:52

Right, another important thing is that, skipping details, you can give a manifold structure to the set of all tangent spaces to a manifold $X$, which is called the tangent bundle of $X$, denoted by $TX$. Now if $f:X\to Y$ is a smooth function between manifolds its differential is a smooth function $TX\to TY$!

Daminark

But good luck!

Manolis Lyviakis

bye!

let me think of it abit

i can make the set of all tangent spaces manifold-like stucture

and now $df$ is a smooth map between the bundles

ok im starting to get it

now the hard part is to do the actual math

Alessandro Codenotti

@ManolisLyviakis as always :P

Manolis Lyviakis

XD

Late347

12:20

is anybody out there?

Alessandro Codenotti

12:36

Well, only got an hour of daylight left. Better get started.

rschwieb

13:50

This is a body out there

quallenjäger

14:25

Suppose I have two infinite dimensional vector space $V$ and $W$ with hamel base $v_j$ and $w_i$. For the tensor product $V \otimes W$, is it true that I can write any element in $V \otimes W$ as finite linear combination of $v_j \otimes w_i$?

Leaky Nun

@quallenjäger yes

quallenjäger

How can I see that this is true?

Leaky Nun

Any element can be written as a finite combination of $v \otimes w$ where $v \in V$ and $w \in W$

any $v$ (resp. $w$) can be written as a finite linear combination of $v_j$ (resp. $w_i$)

small tensor distributes over addition

qed

quallenjäger

Oh yes, thanks

user193319

If $G$ is a topological group, $H$ a closed subgroup, and $G/H$ is given the topology with respect to which the canonical projection $\pi : G \to G/H$ is quotient map, is $\pi$ a closed map?

Nick