- Thread starter
- #1
- Mar 10, 2012
- 835
Hello MHB.
I am not sure what the definition of a diffeomorphism is.
In Spivak's 'Calculus on Manifolds', it is defined as:
A diffeomorphism is a function $h:U\to V$, $U$ and $V$ are open sets in $\mathbb R^n$, such that $h$ is differentiable and has a differentiable inverse.
I have seen the following in Lee's 'Introduction to Smooth Manifolds':
Let $U$ and $V$ be open subsets in $\mathbb R^n$.
A function $F:U\to V$ is smooth if each component of $F$ has continuous partial derivatives of all orders.
A function $h:U\to V$ is a diffeomorphism if it is smooth and has a smooth inverse.
The one in Lee's book is far stronger than the one in Spivak.
I am not sure which one is 'more' standard.
Does this puzzle you too?
I am not sure what the definition of a diffeomorphism is.
In Spivak's 'Calculus on Manifolds', it is defined as:
A diffeomorphism is a function $h:U\to V$, $U$ and $V$ are open sets in $\mathbb R^n$, such that $h$ is differentiable and has a differentiable inverse.
I have seen the following in Lee's 'Introduction to Smooth Manifolds':
Let $U$ and $V$ be open subsets in $\mathbb R^n$.
A function $F:U\to V$ is smooth if each component of $F$ has continuous partial derivatives of all orders.
A function $h:U\to V$ is a diffeomorphism if it is smooth and has a smooth inverse.
The one in Lee's book is far stronger than the one in Spivak.
I am not sure which one is 'more' standard.
Does this puzzle you too?