2007-09-24

Dr. Chen showed me a neat theorem! :)

If I have a smooth (infinitely differentiable) function,

$f:\mathbb R^n \times \{x \ge 0\} \to \mathbb R$

then I can extend it to

$\tilde f:\mathbb R^n \times \mathbb R \to \mathbb R$

such that

$f = \tilde f \big\vert_{\mathbb R^n \times \{x > 0\}}.$

This theorem was stated implicitly in Topology from the Differentiable Viewpoint (as part of a proof that f: M -> N with M a manifold with boundary produces submanifolds of M transversal to the boundary when you take f^-1(n), n ∈ N, if n is regular for M and ∂M), and I have been running around asking professors if they knew the proof of it.

wheeee

2007-09-24

(no subject)