🎉 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
This forum is only accessible to users with an ETH/UZH login.