Hi everyone I'm currently studying Morse Theory on my own and I've come across a proposition where I would need a precise definition of what is meant.
Here is the proposition:
Let $M$ be a manifold that can be embedded as a submanifold into a Euclidean space and let $ f : M → \mathbb{R}$ be a $C_{\infty}$ function. Let $k$ be an integer. Then $f$ and all its derivatives of order $≤ k$ can be uniformly approximated by Morse functions on every compact subset.
The part that confuses me is what is meant by uniform approximation of the derivatives of order ${}$ $ ≤ k$.
An example of a problem I have is with what is meant by derivatives of order bigger than 1. Because from my (very basic) understanding of differential geometry we cannot even generally define the second derivative. It's only well defined on the set of critical points.
I have a lot of other problems with this proposition and I took a look at the proof hoping to find answers to my questions. Sadly the books only proves the case $k=0$ .
What I would need is a very pedantic restatement of the proposition that explains what we mean by approximating higher order derivatives on a compact set and what we even mean by higher order derivatives.
I tried to look on the internet but this exact proposition was nowhere to be found so I really hope someone here knows what it means.