Introduction to Smooth Manifolds 1: Smooth Functions on Euclidean Space

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.

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