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Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts
FROM THE
All legitimate copies of VDGF produced by Princeton University Press are crisply printed on high-quality paper. If you obtain a shoddily printed copy, it's a please return it and purchase a genuine PUP copy.
An inviting, intuitive, and visual exploration of differential geometry and forms
Visual Differential Geometry and Forms fulfills two principal goals. In the first four acts, Tristan Needham puts the geometry back into differential geometry . Using 235 hand-drawn diagrams, Needham deploys Newton's geometrical methods to provide new geometrical explanations of the classical results. In the fifth act, he offers the first undergraduate introduction to Differential Forms that treats advanced topics in an intuitive and geometrical manner.
Unique features of the first four acts four distinct geometrical proofs of the fundamentally important Global Gauss-Bonnet theorem, providing a stunning link between local geometry and global topology; a simple, geometrical proof of Gauss's famous Theorema Egregium; a complete geometrical treatment of the Riemann curvature tensor of an n -manifold; and a detailed geometrical treatment of Einstein's field equation, describing gravity as curved spacetime (General Relativity), together with its implications for gravitational waves, black holes, and cosmology.
The final act provides an intuitive, geometrical introduction to Differential Forms, elucidating such topics as the unification of all the integral theorems of vector calculus; the elegant reformulation of Maxwell's equations of electromagnetism in terms of 2-forms; de Rham cohomology; differential geometry via Cartan's method of moving frames; and the calculation of the Riemann tensor using curvature 2-forms.
Six of the seven chapters of Act V can be read completely independently from the rest of the book, providing a self-contained introduction to Differential Forms.
Requiring only basic calculus and geometry, Visual Differential Geometry and Forms provocatively rethinks the way this important area of mathematics should be understood and taught.
All legitimate copies of VDGF produced by Princeton University Press are crisply printed on high-quality paper. If you obtain a shoddily printed copy, it's a please return it and purchase a genuine PUP copy.
An inviting, intuitive, and visual exploration of differential geometry and forms
Visual Differential Geometry and Forms fulfills two principal goals. In the first four acts, Tristan Needham puts the geometry back into differential geometry . Using 235 hand-drawn diagrams, Needham deploys Newton's geometrical methods to provide new geometrical explanations of the classical results. In the fifth act, he offers the first undergraduate introduction to Differential Forms that treats advanced topics in an intuitive and geometrical manner.
Unique features of the first four acts four distinct geometrical proofs of the fundamentally important Global Gauss-Bonnet theorem, providing a stunning link between local geometry and global topology; a simple, geometrical proof of Gauss's famous Theorema Egregium; a complete geometrical treatment of the Riemann curvature tensor of an n -manifold; and a detailed geometrical treatment of Einstein's field equation, describing gravity as curved spacetime (General Relativity), together with its implications for gravitational waves, black holes, and cosmology.
The final act provides an intuitive, geometrical introduction to Differential Forms, elucidating such topics as the unification of all the integral theorems of vector calculus; the elegant reformulation of Maxwell's equations of electromagnetism in terms of 2-forms; de Rham cohomology; differential geometry via Cartan's method of moving frames; and the calculation of the Riemann tensor using curvature 2-forms.
Six of the seven chapters of Act V can be read completely independently from the rest of the book, providing a self-contained introduction to Differential Forms.
Requiring only basic calculus and geometry, Visual Differential Geometry and Forms provocatively rethinks the way this important area of mathematics should be understood and taught.
530 pages, Hardcover
Published July 13, 2021
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377 people want to read
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Displaying 1 - 7 of 7 reviews
September 10, 2021
Part 5 (the only one I've read) is far and away the best intuitive explanation of differential forms I've come across. Like Visual Complex Analysis I wouldn't recommend reading this without a more formal book as a companion.
Parts 1-4 are Riemannian geometry, I probably won't read them.
Maybe worth noting that the prose is overwrought and kind of annoying, but it basically only shows up in the intro sections so whatever.
Update: I was not nearly positive enough in my initial review. This is getting added to my list of best math textbooks ever. Still needs a formal book to go with it though.
Update 2: I glanced through the other 4 sections. Overall they felt much less valuable, probably because the topics covered already have easily accessible geometric intuitions. I don't think getting an even more visual than usual explanation of curvature is very helpful, especially when it comes at the cost of rigor.
Parts 1-4 are Riemannian geometry, I probably won't read them.
Maybe worth noting that the prose is overwrought and kind of annoying, but it basically only shows up in the intro sections so whatever.
Update: I was not nearly positive enough in my initial review. This is getting added to my list of best math textbooks ever. Still needs a formal book to go with it though.
Update 2: I glanced through the other 4 sections. Overall they felt much less valuable, probably because the topics covered already have easily accessible geometric intuitions. I don't think getting an even more visual than usual explanation of curvature is very helpful, especially when it comes at the cost of rigor.
June 26, 2022
As others on our site here have commented correctly, this text needs to be read after, and along with, more formal, topical discussions. That having been said, with the appropriate background and reference aids at hand, the reader will be taken through a well written and, at times, extraordinarily clear presentation.
October 30, 2022
A delightful little textbook for differential geometry
Differential geometry done through deep intuition guiding pictures. I really enjoyed it, and it feels like a lot of it bedded down quite deeply in my mind. Sometimes lacked in the clarity of arguments you expect from a maths textbook (I didn't quite know what had been shown using what, for example), but broadly excellent!
Good if you want to understand, rather than just serve as a reference of truth.
Differential geometry done through deep intuition guiding pictures. I really enjoyed it, and it feels like a lot of it bedded down quite deeply in my mind. Sometimes lacked in the clarity of arguments you expect from a maths textbook (I didn't quite know what had been shown using what, for example), but broadly excellent!
Good if you want to understand, rather than just serve as a reference of truth.
February 14, 2025
I stumbled upon this book almost by accident, and now I have fallen head over heals in love with differential geometry. Needham's geometric approach to explaining and proving astounding mathematic theorems is both intuitive and satisfying and is sequenced such that every new discovery is both obvious and exciting. Needham's commentary is gripping while still lighthearted and even humorous at times. My biggest critiques are that I wish we spent more time with higher-dimensional manifolds and Act V about differential forms felt more like an afterthought to me than the jaw-dropping conclusion it was established as. However, I think my lack of experience studying this field played a large role in these critiques, and the sheer sense of wonder this book brought me more than made up for them.
September 2, 2023
Good source for intuition when needed.
April 17, 2022
I enjoyed the first three sections of the book immensely. As someone for whom geometric insight is lacking, I thought that the book did an excellent job helping me to build the tools to perform geometric analysis. Part 4 was inaccessible to me, as I was not patient enough to sit with a companion text to help me better understand the underlying concepts and new notation schema.
I will likely go through this book again, and I am sure I will pick up the Visual Complex Analysis book and work through that (though, admittedly, not for some time).
I will likely go through this book again, and I am sure I will pick up the Visual Complex Analysis book and work through that (though, admittedly, not for some time).
March 12, 2023
Loved this book. Read it in between jobs in 2021. Part of my nightly read before bed. Got a better understanding of curvature, hyperbolic geometry, and forms.
Displaying 1 - 7 of 7 reviews