About Arnold's ODE Book Notation

In summary: So what you are looking for is a function from the set of all maps in ##X## to the set of all maps in ##Y##, which is just what ##T_g## defines.It currently is bog-standard notation used in an almost uncountable number of books on my shelves.But if T takes m to m, how is the same which T takes g of G(or gh of G) to S(M)=group of all bijective transformations of M. That is correct. If ##T## takes ##g## to ##S\left(M\right)##, then ##T_g## is in ##S\left(M\right)##.
  • #1
Martin T
12
0
In Arnold's book, ordinary differential equations 3rd. WHY Arnold say Tg:M→M instead of Tg:G→S(M) for transformations Tfg=Tf Tg,
Tg^-1=(Tg)^-1.

Let M be a group and M a set. We say that an action of the group G on the set M is defined if to each element g of G there corresponds a transformation Tg : M→M of the set M, to the product and inverse elements corresponds Tfg=TfTg, Tg^-1=(Tg)^-1
 
Physics news on Phys.org
  • #2
You know that you only address members who a) have this book and b) are willing to take it from the shelf?
 
  • #3
Yes, but Arnold is popular and this paragraph is being kill me.
 
  • #4
As you wish. I could probably answer your question if I only had the book or you had put a little effort in describing the situation. I prefer to read books written in my own language. So, good luck! I'm out.
 
  • #5
My guess is that ##T : G \rightarrow S\left(M\right)## with ##g \in G \mapsto T_g \in S\left(M\right)##.
 
  • Like
Likes mathwonk
  • #6
Here the Page, the Last paragragh, thanks.
 

Attachments

  • NuevoDocumento 2018-08-23_1.jpg
    NuevoDocumento 2018-08-23_1.jpg
    21.4 KB · Views: 483
  • #7
George Jones said:
My guess is that ##T : G \rightarrow S\left(M\right)## with ##g \in G \mapsto T_g \in S\left(M\right)##.
Yes I think the same, then Arnold is wrong with
Tg:M→M.
 
  • #8
Martin T said:
Yes I think the same, then Arnold is wrong with
Tg:M→M.

If what I wrote is correct, then Arnold is correct., i.e., I, in part, wrote ##T_g \in S\left(M\right)##. This means that ##T_g : M \rightarrow M##. This is fairly common notation.
 
  • Like
Likes Martin T
  • #9
George Jones said:
If what I wrote is correct, then Arnold is correct., i.e., I, in part, wrote ##T_g \in S\left(M\right)##. This means that ##T_g : M \rightarrow M##. This is fairly common notation.
Then it is a old fashioned notation, thanks
 
  • #10
Martin T said:
Then it is a old fashioned notation, thanks

It currently is bog-standard notation used in an almost uncountable number of books on my shelves.
 
  • Like
Likes Martin T
  • #11
But if T takes m to m, how is the same which T takes g of G(or gh of G) to S(M)=group of all bijective transformations of M.
And say Arnold's books are ""pedagogic".
 
  • #12
Martin T said:
And say Arnold's books are ""pedagogic".

Put an end to your sarcastic comments, or I will put an end to helping you.

Martin T said:
But if T takes m to m

This is not what I wrote. I wrote

George Jones said:
##T : G \rightarrow S\left(M\right)##

In other words, ##T## takes ##G## to ##S\left(M\right)##. Consequently, using functional bracket notation, ##T\left(g\right)## is in ##S\left(M\right)##, i.e., ##T\left(g\right)## takes ##M## to ##M##. Here, bracket notation becomes too cluttered/confusing, so it is conventional to denote ##T\left(g\right)## as ##T_g##.
 
  • Like
Likes Martin T
  • #13
Yes, I understand, but in the book Arnold first defines an action:
Let M be a group and M a set. We say that an action of the group G on the set M is defined if to each element g of G there corresponds a transformation Tg : M→M of the set M, to the product and inverse elements corresponds Tfg=TfTg, Tg^-1=(Tg)^-1

Why? It is correct(Arnold talk about homomorphims after)?
 
  • #14
Saw
 

Attachments

  • Ar-EDO_62-62.pdf
    66.8 KB · Views: 305
  • Ar-EDO_63-63.pdf
    51.6 KB · Views: 302
  • #15
Martin T said:
Yes, I understand, but in the book Arnold first defines an action:
Let M be a group and M a set. We say that an action of the group G on the set M is defined if to each element g of G there corresponds a transformation Tg : M→M of the set M, to the product and inverse elements corresponds Tfg=TfTg, Tg^-1=(Tg)^-1

Why? It is correct(Arnold talk about homomorphims after)?
It is all correct and standard. Every algebra book (any maths book really) that defines action of a group on a set uses these notation.

Probably what confuses you is that you have a map ##f:A\rightarrow B##, where the target set ##B## is a set of maps, say between the sets ##X## and ##Y##, so the image of any element ##a\in A## is map itself ##f(a) : X \rightarrow Y##.
 
  • Like
Likes Martin T

1. What is the purpose of Arnold's ODE Book Notation?

The purpose of Arnold's ODE Book Notation is to provide a standardized way of writing and solving ordinary differential equations (ODEs). It simplifies and streamlines the notation used in ODEs, making it easier for scientists and mathematicians to communicate and understand the equations.

2. Who created Arnold's ODE Book Notation?

Arnold's ODE Book Notation was created by mathematician and physicist Vladimir Arnold. He published it in his book "Ordinary Differential Equations" in 1992.

3. How is Arnold's ODE Book Notation different from other notations for ODEs?

Arnold's ODE Book Notation is unique in that it uses a single-letter notation for the independent variable (usually denoted as t), making it easier to write and read equations. It also uses bold letters for vectors and matrices, making them stand out and reducing the chance of errors in equations.

4. Can Arnold's ODE Book Notation be used for all types of ODEs?

Yes, Arnold's ODE Book Notation can be used for all types of ODEs, including first-order, second-order, and higher-order equations. It can also be applied to systems of ODEs.

5. Is Arnold's ODE Book Notation widely used in scientific research?

Yes, Arnold's ODE Book Notation is widely used in scientific research, particularly in the fields of mathematics, physics, and engineering. It has become a standard notation for writing and solving ODEs and is taught in many university courses.

Similar threads

  • Linear and Abstract Algebra
Replies
4
Views
1K
  • Science and Math Textbooks
Replies
8
Views
2K
  • Linear and Abstract Algebra
Replies
1
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
466
  • Science and Math Textbooks
Replies
5
Views
2K
  • Differential Equations
Replies
5
Views
608
Replies
2
Views
2K
  • Linear and Abstract Algebra
Replies
4
Views
2K
  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Science and Math Textbooks
Replies
9
Views
3K
Back
Top