Undergrad Do diffeomorphisms have to be one-to-one functions?

Click For Summary
A diffeomorphism requires a differentiable inverse, which necessitates that the original function be one-to-one to ensure the inverse is also a function. If a function is not one-to-one, it cannot have an inverse that qualifies as a function. Therefore, the original function must be injective for it to be classified as a diffeomorphism. This concept is fundamental in understanding the properties of functions in calculus. The discussion highlights the importance of injectivity in the context of diffeomorphisms.
Antineutrino
Messages
6
Reaction score
0
The definition of a diffeomorphism involves the differentiable inverse of a function, so must the original function be one-to-one to make its inverse a function, or can the inverse be a relation and not a function?

Sorry if it's a silly question, I am just a second semester calc student who looked at this for fun.
 
Physics news on Phys.org
Antineutrino said:
The definition of a diffeomorphism involves the differentiable inverse of a function, so must the original function be one-to-one to make its inverse a function, or can the inverse be a relation and not a function?

Sorry if it's a silly question, I am just a second semester calc student who looked at this for fun.
If a function is not one-to-one, then it has no inverse.
 

Similar threads

Diffeomorphisms and active transformations
  • · Replies 4 ·
Replies
4
Views
2K
How to show that a locally diffeomorphic mapping from R to R is open?
  • · Replies 3 ·
Replies
3
Views
6K
Differentiable structures and diffeomorphisms
  • · Replies 5 ·
Replies
5
Views
5K
Graduate Diffeomorphisms & the Lie derivative
  • · Replies 9 ·
Replies
9
Views
4K
Undergrad What Are the Axioms of Fuzzy Logic and How Do They Extend Boolean Algebra?
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Collapse of particle wave function and single Universe wave function?
  • · Replies 43 ·
2
Replies
43
Views
5K
Challenge Math Challenge Thread (October 2023)
  • · Replies 25 ·
Replies
25
Views
5K
Graduate Recover Hamilton equation from 2-form defined on phase space
  • · Replies 3 ·
Replies
3
Views
2K
Integral curves and one-parameter groups of diffeomorphisms
  • · Replies 2 ·
Replies
2
Views
6K
I want to see the plot of an inverse function -- can I use a TI84?
  • · Replies 10 ·
Replies
10
Views
6K