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 in coordinates with superscripts . I prefer subscripts and slice notation: .
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 as , and the standard coordinates on as .
In case you didn’t know that. And yet are reading a book about smooth manifolds.