Description
The aim of this course is to give a modern and rigorous account of advanced multivariate calculus. The spaces on which we set up our differentiation and integration are called manifolds.
Basically they are pieces of n-dimensional space stitched together like a soccer ball or a T-shirt or the universe is made up from small pieces of fabric. An important feature of manifolds is that there is usually no preferred system of coordinates so that one has to learn to be flexible in changing coordinates as needed.
This course provides essential background for master level courses on (algebraic) geometry, topology, (partial) differential equations. More specifically we will introduce tensors as sections of vector bundles. In particular differential forms and their fundamental theorem of calculus (Stokes). We will also make small excursions into complex algebraic curves, Lie groups and Hamiltonian dynamics.
Lectures and examination
Weekly lectures and problem sessions. Written exam plus weekly homework. The final grade is a weighted average of homework and the final test. The homework counts as the area of a disk with radius a quarter in the hyperbolic plane.
Prerequisites
Linear algebra 1,2, algebra 1, analyse 1,2,3,4.
Literature
Course notes available from the webpage