Turn on thread page Beta
    • Thread Starter
    Offline

    1
    ReputationRep:
    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!
    • PS Helper
    Offline

    14
    PS Helper
    A diffeomorphism is a smooth map with a smooth inverse. So say f : U \to V, where U,V are (simply connected) open subsets of \mathbb{R}^m, \mathbb{R}^n respectively (or more generally if they're manifolds). Then f is a diffeomorphism if:
    (a) f is bijective
    (b) f has derivatives of all orders
    (c) f^{-1} has derivatives of all orders
    Simples
    Offline

    2
    ReputationRep:
    (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!
     f : U \rightarrow V , 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
    • Thread Starter
    Offline

    1
    ReputationRep:
    thanks guys
    • Thread Starter
    Offline

    1
    ReputationRep:
    Quick question. Why does it need to be bijective. Why not just injective??

    Thanks
    Offline

    6
    ReputationRep:
    (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?
    • Thread Starter
    Offline

    1
    ReputationRep:
    (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
    Offline

    6
    ReputationRep:
    (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.
    • Thread Starter
    Offline

    1
    ReputationRep:
    (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
    Offline

    10
    ReputationRep:
    (Original post by thebadgeroverlord)
     f : U \rightarrow V , 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 C^1 manifolds or C^{\infty} manifolds...
 
 
 
The home of Results and Clearing

1,274

people online now

1,567,000

students helped last year

University open days

  1. London Metropolitan University
    Undergraduate Open Day Undergraduate
    Sat, 18 Aug '18
  2. Edge Hill University
    All Faculties Undergraduate
    Sat, 18 Aug '18
  3. Bournemouth University
    Clearing Open Day Undergraduate
    Sat, 18 Aug '18
Poll
A-level students - how do you feel about your results?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

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

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.