# 1. Smooth Manifolds

### đ Welcome to Differential Geometry I đ

Calculus (the word comes from the Latin and means âsmall pebbleâ) is a method of measuring the rate at which things change. It is absolutely crucial to all applications of mathematics, and is arguably the single most important concept in theoretical physics. Ever since you were very young (Kindergarten, Basisjahr, etc), youâve known how âdo calculusââthat is, differentiation and integrationâon Euclidean spaces.

Unfortunately, many interesting mathematical systems are not defined on (open sets of) Euclidean spaces. And on an arbitrary metric or topological space, the type of calculus you know and love does not make sense. Indeed, the differential of a function $f$ at a point $x$ can be thought of as the âbest linear approximationâ to $f$ near $x$, and the word âlinearâ simply has no meaning on a general topological space.

At its heart, differential geometry is the study of smooth manifolds, which are a class of topological spaces for which it does make sense to differentiate (and later, integrate) things on. Todayâs lecture is all about the definition of a smooth manifold.

##### Remarks

- The lecture notes will typically contain more material than I have time to cover in lecture. Anything marked with a $(\clubsuit)$ is non-examinable and you are welcome to ignore it.
- Itâs far too early to talk about this now, but just so you know: the Differential Geometry I (and II) exam will be an oral exam. I will only test material from the Lecture Notes and the Problem Sheets. (There will be no âunseenâ problems.)
- I encourage all students who are taking the course to join my forum (ETH/UZH login only, sorry). After joining you are asked to choose a username, and I am more than happy for you to choose something anonymous. Your email address is never shown publicly.

I will post a âQ&Aâ thread there after each lecture (the button below is a direct link to todayâs thread). Please do ask questions if there is anything you donât understand, or if you spot typos in my lecture notes (and trust me, there will be LOTS of typosâŚ ) I will try to answer every single one đ

More information about prerequisites, recommended textbooks, the exam, and a collection of bad math jokes are also available on the forum.

Comments and questions?