# 1. Smooth Manifolds

### 🎉 Welcome to Differential Geometry I 🎉

Calculus (the word comes from the Latin and means “small pebble”) 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), you’ve known how “do calculus” – that is, differentiation and integration – on 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 $p$ can be thought of as the “best linear approximation” to $f$ near $p$, and the word “linear” 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. Today’s lecture is all about the definition of a smooth manifold.

##### Remarks

- The Zoom lecture recording is available on the DMATH forum.

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- The lecture notes for the 2018-2019 version of the course I taught are also available as a single PDF on the forum. At least to begin with, the content of the 2020-2021 version of the course will be broadly similar.
- The lecture notes will typically contain more material than I have time to cover in lecture. At the end of the lecture there is a
**Bonus Material**section — all of this is non-examinable.

I will post a “Q&A” thread there after each lecture (the button below is a direct link to today’s thread). Please do ask questions if there is anything you don’t understand, or if you spot typos in my lecture notes (and trust me, there will be LOTS of typos… ) I will try to answer every single one 😀

Comments and questions?