**Description**

In this course we sharpen and combine our tools from linear algebra and calculus to address geometric questions such as:

What is the shortest path from A to B in a curved space? What do we mean by ‘curved’ anyway? How can we understand vector fields qualitatively? And what does the graph of a holomorphic function look like in R^4?

A natural setting for addressing such questions are manifolds: spaces that locally look like R^n.

We will clean up the the vector calculus of R^n and lift it to the context of manifolds using differential forms. This includes a vast generalization of the fundamental theorem of calculus: the general Stokes theorem.

In the setting of manifolds we will be able to prove the famous Gauss-Bonnet theorem:

**The integral of the curvature of a surface equals the total index of any vector field on the surface and this in turn equals the Euler characteristic.**

This illustrates the beautiful interaction of geometry, differential equations and topology that manifolds are all about.

**Audience**

This course is aimed at a broad audience of 3rd year bachelor students, particularly all students with an interest in geometry and analysis. By working concretely in R^n we aim to strike a balance between learning how to prove theorems and how to do computations in curved space.

**Examination**

Written exam plus weekly homework

**Prerequisites**

Linear Algebra 1,2, Analysis 2,3, Complex Analysis, Algebra 1. Topology is not a prerequisite.

**Link**

Course homepage:“http://www.rolandvdv.nl/M15/”

**Literature**

Course notes available from the homepage.