A New Approach to Differential Geometry using Clifford's Geometric Algebra

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Differential geometry is the study of the curvature and calculus of curves and surfaces. A New Approach to Differential Geometry using Clifford's Geometric Algebra simplifies the discussion to an accessible level of differential geometry by introducing Clifford algebra. This presentation is relevant because Clifford algebra is an effective tool for dealing with the rotations intrinsic to the study of curved space.

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Complete with chapter-by-chapter exercises, an overview of general relativity, and brief biographies of historical figures, this comprehensive textbook presents a valuable introduction to differential geometry. It will serve as a useful resource for upper-level undergraduates,  beginning-level graduate students, and researchers in the algebra and physics communities.

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A New Approach to Differential Geometry using Clifford's Geometric Algebra Review

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About half the sections are marked with asterisks to let you know future materia is not dependent on it . I just finished reading all the 'unstarred' material. I'll update this review after I go back and read the rest, but I wanted to get a review up because there is no feedback for the book so far.

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I have wanted to learn differential geometry for a long time. I tried the differential forms approach first (recently), but it was extremely slow-going, I was afraid I was drawing a lot of wrong conclusions, and my brain was constantly hurting. I discovered this book only 2 or 3 of weeks ago - I hadn't even heard of geometric algebra before then. I had intended to do a thorough study of topology and then give forms another go, but this sounded intriguing so I decided to read this book first.

I don't think I've got as much out of a book since I read Spivak's "Calculus", and I don't think I've enjoyed reading a book as much since I read "The Lord of the Rings". "A new approach..." reads like a novel - it's hard to put it down. I haven't really gotten stuck on anything - I'm not sure if the author is amazing at explaining things or geometric algebra is extremely intuitive, but this book just makes sense.

Here are some notes about the book you might be interested in if you are considering it:

-As far as prerequisites go, you are definitely prepared if you have had calc 1 - 3 and linear algebra (the prerequisites listed by the author). Calc 3 is definitely required, but you would probably be fine without having taken a linear algebra course - as long as you know how to transpose/invert/multiply/add matrices, know what linearly dependent/independent mean, and know the definition of the determinant using permutations of the elements (as well as how swapping a pair of rows/columns of a matrix, transposing the matrix, etc. will effect its determinant). The author mentions in the introduction that differential equations do not enter into the problem sets, but I can think of 2 or 3 examples where they did (at least the way I solved the problems).

-The book is very self-contained. It was nice to not have to constantly look up terminology on Google, since everything was properly defined in the book.

-There are an abundance of figures, which is a great thing for a geometry book. I wish more authors understood how much a picture helps sometimes.

-There are no solutions to the exercises. I know this bothers some people, but 90% of the exercises are the type of questions where you know if you got it right or not anyways. For the other 10% - If the exercise requires you to calculate something explicit, like the components of the metric tensor for a given surface, there's almost always something to compare your answer to in the previous section or in a follow-up problem.

-The author likes to include historical sections at the end of chapters. These are mostly mini-biographies of important mathematician/scientists who contributed a lot to what you read about in that chapter. They concentrate on their subjects' lives and personalities more than the their influence on the math. I personally enjoyed reading them a lot, but they are "starred" so you can skip them if you want without really missing anything.

-The author points out similarities with differential forms when relevant. I've started to read some differential forms material again, so I appreciate that he did that. After reading the 'non-starred' sections of this book, forms are making a lot more sense to me. In my case, it was much easier to learn the concepts using geometric algebra - not to mention that, so far, it also appears to be easier/less tedious to work with.

The author really did a great job with this book. He wrote something that is enjoyable to read, accessible to non-math-major undergraduate students, derives important results in differential geometry without using complicated differential forms, and brought me - in less than three weeks - from being completely inexperienced to being able to follow along with and understand the author's calculation of the orbit of Mercury using general relativity. If you are interested in learning differential geometry, I say this is a great place to start.

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