Working on Introduction to Smooth Manifolds by Loring Tu (AKA $$L^2$$) with Charles.

## 1.1: Smooth vs. Analytic functions

I don’t yet see why this distinction is the first thing the book talks about.

Smooth functions are infinitely differentiable. Analytic functions are even stronger since they equal their Taylor series.

I already hate the convention of writing a point $$p$$ in coordinates with superscripts $$(p^1 \dots p^n)$$. I prefer subscripts and slice notation: $$p := p_{1:n}$$.

Bump functions are cool, since you can create a smooth function that’s totally flat outside an arbitrarily small region.

Charles pointed out that non-analytic smooth functions are physically because they change their values in a finite amount of time but can have 0 derivative of every order.

## 1.2: Taylor’s Theorem with Remainder

Smooth functions can be approximated well by a Taylor series.

This section basically derives the Taylor formula.

Here’s the ending note:

NOTATION. It is customary to write the standard coordinates on $$\mathbb{R}^2$$ as $$x, y$$, and the standard coordinates on $$\mathbb{R}^3$$ as $$x, y, z$$.

In case you didn’t know that. And yet are reading a book about smooth manifolds.