In this lecture we introduce the notion of a singular cube in a smooth manifold, and explain how to integrate a differential form over a singular cube. We then define a singular chain as a formal sum of singular cubes, and prove a local version of Stokes’ Theorem.

Next lecture we will prove the global Stokes’ Theorem.

##### Remark

The final lecture of the semester will be completely non-examinable. In that lecture we will give a proof of the de Rham Theorem, which tells us that de Rham cohomology is isomorphic to singular cohomology. This lecture will only be understandable to students who are already familiar with basic algebraic topology. (But don’t worry if you are not—you can just start your winter vacation early 😀).