Lecture Notes on Differential Forms

Abstract

This is a series of lecture notes, with embedded problems, aimed at students studying differential topology. Many revered texts, such as Spivak's "Calculus on Manifolds" and Guillemin and Pollack's "Differential Topology" introduce forms by first working through properties of alternating tensors. Unfortunately, many students get bogged down with the whole notion of tensors and never get to the punch lines: Stokes' Theorem, de Rham cohomology, Poincare duality, and the realization of various topological invariants (e.g. the degree of a map) via forms, none of which actually require tensors to make sense! In these notes, we'll follow a different approach, following the philosophy of Amy's Ice Cream: Life is uncertain. Eat dessert first. We're first going to define forms on $\mathbb{R}^n$ via unmotivated formulas, develop some proficiency with calculation, show that forms behave nicely under changes of coordinates, and prove Stokes' Theorem. This approach has the disadvantage that it doesn't develop deep intuition, but the strong advantage that the key properties of forms emerge quickly and cleanly. Only then, in Chapter 3, will we go back and show that tensors with certain (anti)symmetry properties have the exact same properties as the formal objects that we studied in Chapters 1 and 2. This allows us to re-interpret all of our old results from a more modern perspective, and move onwards to using forms to do topology.