A Beginner's Guide to Differential Forms

by Peter Collier

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

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.By the author of A Most Incomprehensible Notes Towards a Very Gentle Introduction to the Mathematics of Relativity (ISBN 9780957389465) and Movement of the Basic Astronomical Calculations Explained With 29 Spreadsheets (ISBN 9780957389489).

Genres: Mathematics

146 pages, Kindle Edition

Published July 1, 2021

About the author

Founder of Encounter Books in California, Collier was publisher from 1998-2005. He co-founded the Center for the Study of Popular Culture with David Horowitz. Collier wrote many books and articles with Horowitz. Collier worked on the website FrontpageMag. He was an organizer of Second Thoughts conferences for leftists who have moved right.

Librarian Note: There is more than one author in the Goodreads database with this name.

Reviews

[Jane · Rating 5 out of 5]

A great aid in learning applied mechanics

The mathematical signs, symbols, and operations are defined in simple language with useful diagrams. The pace is defined by chapters— just be sure to understand each one and help yourself by returning to earlier chapters whenever the going gets fuzzy. The references will be important should you go further; if I had Collier’s book when I first read many of them I would be farther ahead.

[Des Bladet · Rating 3 out of 5]

The last chapter has some cool topological theorems, including the famous Hairy Ball Theorem. I liked that chapter. The other chapters I didn't much care for, but you do get an account of how differential forms work, at a fairly mechanical level. The generalized Stokes theorem is exhibited, but not proved, and mostly put to use to derive theorems of vector calculus you probably already know, and it is not obvious what the point of that is supposed to be. The treatment of Maxwell's Equations is much the same: you get a new set of notation, but no new insights.

[Alex Siryj · Rating 5 out of 5]

Good intro to differential forms

This is a very accessible book for learning differential forms. It is informal compared to most math texts. Also in contrast to a lot of math texts, the author doesn't waste any time and gets straight to the point. I really liked this book! I'd recommend it to anyone interested in modern calculus.