Ebook: A Beginner's Guide to Differential Forms

Differential forms eat tangent vectors, spit out numbers and do this in an alternating multilinear fashion. These properties make differential forms an essential tool for doing calculus on manifolds and thus of great interest to mathematicians and physicists. This book is aimed at the general reader who is interested in tackling a relaxed but wide-ranging introduction to this fascinating subject. The only prerequisite is a reasonable foundation in advanced, school-level mathematics. The text includes numerous worked problems. Topics covered include:

• An introduction to basic concepts.
• How differential forms eat tangent vectors and spit out numbers.
• Manipulating differential forms, including the wedge product, differentiation and integration.
• How differential forms provide an alternative means of understanding three-dimensional vector calculus.
• The generalised Stoke's theorem, which applies to manifolds of any dimension.
• Maxwell's equations in the language of differential forms.
• Using differential forms to prove three nice topological theorems, including the famous hairy ball theorem.