Name: Functional Differential Geometry
ISBN: 9780262019347

Functional Differential Geometry

An explanation of the mathematics needed as a foundation for a deep understanding of general relativity or quantum field theory.Physics is naturally expressed in mathematical language. Students new to the subject must simultaneously learn an idiomatic mathematical language and the content that is expressed in that language. It is as if they were asked to read Les Misérables while struggling with French grammar. This book offers an innovative way to learn the differential geometry needed as a foundation for a deep understanding of general relativity or quantum field theory as taught at the college level.The approach taken by the authors (and used in their classes at MIT for many years) differs from the conventional one in several ways, including an emphasis on the development of the covariant derivative and an avoidance of the use of traditional index notation for tensors in favor of a semantically richer language of vector fields and differential forms. But the biggest single difference is the authors' integration of computer programming into their explanations. By programming a computer to interpret a formula, the student soon learns whether or not a formula is correct. Students are led to improve their program, and as a result improve their understanding.

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michaelsbradleyApr 2025

Or, if you want to learn the core ideas and math involved, but in terms of computer programming, see:

Functional Differential Geometry by Gerald Sussman, same author behind Structure and Interpretation of Computer Programs

https://archive.org/details/FunctionalDifferentialGeometry/

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Y_YDec 2024

If you liked the "functional" style of calculus in SICM, or want a calculus only book in this vein I recommend Baby Spivak: https://en.wikipedia.org/wiki/Calculus_on_Manifolds_(book)

(And obviously Functional Differential Geometry by the authors of SICM)

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PheonixPhartsMay 2022

I'm a huge fan of that book, but am very glad you called it out as more people should know about it. there's also a great companion book "Functional Differential Geometry".

My one caveat with that approach is it requires a very specific implementation of their library in MIT Scheme. There probably is a racket implementation by now, but it still looks like it's most small personal github projects.

The automatic differentiation landscape has changed considerably since that book was written. I would love to eventually be able to put together something like a guide to Lagrangian mechanics with JAX.

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enriqutoMay 2020

If you are more mathematically inclined, there's also "Functional Differential Geometry" by the same authors, and in the same spirit, but much shorter. The goal of the book is essentially to write the Euler-Lagrange equations in lisp, which is a breathtakingly beautiful thing.

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kachnuv_ocasekMay 2020

On a related note, and seeing that some people around here are interested to learn the topic from different angle: Gerald Jay Sussman (among others, co-author of Structure and Interpretation of Computer Programs) has co-written a book on differential geometry called Functional Differential Geometry.

It builds up the theory of differential geometry using Scheme, just like SICP and SICM, and is a fantastic read for programmers wanting to grasp the topic in a more familiar language.

It is available for download for free on the publisher's website: https://mitpress.mit.edu/books/functional-differential-geome...

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KoshkinJan 2017

You can try Functional Differential Geometry by Sussman and Wisdom (which exists both as a printed book and as a freely downloadable PDF file). It is an interesting experiment in teaching mathematics by heavily relying on the computer for doing symbolic transformations, which requires the complete clarity in the notation and the understanding of the material.

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