Course Description

In this course, we will learn how to measure the shape of a space by means of calculus. We will start by introducing differentiation and integration in Euclidean space of high dimensions. Then we will understand how to talk about curves and surfaces rigorously, and how to measure their curvatures by differentiation. After these preparations we will come to the core part, namely the Fundamental Theorem of Calculus. The Theorem leads to the birth of several foundational subjects such as residue theory, de Rham cohomology and characteristic classes. Finally we will study the Gauss-Bonnet Theorem (for surfaces), which is a deep discovery on the hidden relation between the local geometry and the global topology of a space.