Difference between revisions of "Smooth function"

Latest revision as of 16:16, 14 April 2015

Definition

A smooth function on a smooth $n$-manifold, $(M,\mathcal{A})$, is a function^[1] $f:M\rightarrow\mathbb{R}$ that satisfies:

• $\forall p\in M\ \exists\ (U,\varphi)\in\mathcal{A}[p\in U\wedge f\circ\varphi^{-1}:\varphi(U)\subseteq\mathbb{R}^n\rightarrow\mathbb{R}\in C^\infty]$
  • That is to say $f\circ\varphi^{-1}$ is smooth in the usual sense - of having continuous partial derivatives of all orders.

Extending to vectors

Note that given an $f:M\rightarrow\mathbb{R}^k$ this is actually just a set of functions, $f_1,\cdots,f_k$ where $f_i:M\rightarrow\mathbb{R}$ and $f(p)=(f_1(p),\cdots,f_k(p))$

We can define $f:M\rightarrow\mathbb{R}^k$ as being smooth $\iff\forall i=1,\cdots k$ we have $f_i:M\rightarrow\mathbb{R}$ being smooth

Notations

The set of all smooth functions

Without knowledge of smooth manifolds we may already define $C^\infty(\mathbb{R}^n)$ - the set of all functions with continuous partial derivatives of all orders.

However with this definition of a smooth function we may go further:

The set of all smooth functions on a manifold

Given a smooth $n$-manifold, $M$, we now know what it means for a function to be smooth on it, so:

Let $$f\in C^\infty(M)\iff f:M\rightarrow\mathbb{R}$$ is smooth

See also

• Smooth map
• Smooth manifold
• Smoothly compatible charts

References

1. Jump up ↑ Introduction to smooth manifolds - John M Lee - Second Edition