Turn on thread page Beta
 You are Here: Home >< Maths

# Diffeomorphisms watch

1. Does anyone have a set of things you need to check to know if a mapping is a diffeomorphism??

Such as a set of axioms or something. I can find them. I have some in my notes, but I honestly do not trust this lecturer. He was rubbish!

Thanks!
2. A diffeomorphism is a smooth map with a smooth inverse. So say , where are (simply connected) open subsets of respectively (or more generally if they're manifolds). Then is a diffeomorphism if:
(a) is bijective
(b) has derivatives of all orders
(c) has derivatives of all orders
Simples
3. (Original post by adie_raz)
Does anyone have a set of things you need to check to know if a mapping is a diffeomorphism??

Such as a set of axioms or something. I can find them. I have some in my notes, but I honestly do not trust this lecturer. He was rubbish!

Thanks!
, where U and V are open, is a diffeomorphism if f is bijective and both f and f^(-1) are continuous with first order partial derivitives
4. thanks guys
5. Quick question. Why does it need to be bijective. Why not just injective??

Thanks
6. (Original post by adie_raz)
Quick question. Why does it need to be bijective. Why not just injective??

Thanks
What's the relationship between a function f being a bijection and f having an inverse?
7. (Original post by Daniel Freedman)
What's the relationship between a function f being a bijection and f having an inverse?
I realise that for any function with a codomain the function must be a bijection to have an inverse (wiki explained that).

Does that mean that now every function has a codomain? If not what does it have instead? Just an image?

You are being a great help. Thanks
8. (Original post by adie_raz)
I realise that for any function with a codomain the function must be a bijection to have an inverse (wiki explained that).

Does that mean that now every function has a codomain? If not what does it have instead? Just an image?

You are being a great help. Thanks
Every function has a domain and codomain (and an image). When you define a specific function, you specify it's domain and codomain.
9. (Original post by Daniel Freedman)
Every function has a domain and codomain (and an image). When you define a specific function, you specify it's domain and codomain.
Brilliant thanks
10. (Original post by thebadgeroverlord)
, where U and V are open, is a diffeomorphism if f is bijective and both f and f^(-1) are continuous with first order partial derivitives
That's not the meaning of diffeomorphism I'm familiar with, but I suppose it depends on whether you're working with manifolds or manifolds...

Turn on thread page Beta

### Related university courses

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: April 7, 2011
The home of Results and Clearing

### 1,274

people online now

### 1,567,000

students helped last year
Today on TSR

### IT'S TODAY!

A-level results chat here

### University open days

1. London Metropolitan University
Sat, 18 Aug '18
2. Edge Hill University
Sat, 18 Aug '18
3. Bournemouth University
Clearing Open Day Undergraduate
Sat, 18 Aug '18
Poll
Useful resources

## Make your revision easier

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams

Can you help? Study help unanswered threads

## Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE