数学2020math.ucla.edu
微分形式与积分
Differential Forms and Integration (2008)
进阶 · 约 90 分钟读完前置知识：多元微积分、线性代数、基础拓扑
导读
这是 UCLA 数学家陶哲轩（Terence Tao）2008 年的讲义，介绍微分形式与积分的关系，涵盖如何用微分形式统一表述多元微积分中的积分概念（如斯托克斯定理等）。文中从微分形式在 R^n 上的定义讲起，逐步推广到流形上的积分。
HN 关注点：微分形式是许多数学和物理学习者难以建立直觉的主题，HN 读者借此讨论如何真正理解它以及推荐学习资源。
HN 评论精选
这几条评论里：质疑 1 补充 2 亲历 1
补充推荐用 Scheme 讲微分几何的免费书籍，适合程序员
顺带一提，SICP 的合著者 Gerald Jay Sussman 等人写了一本微分几何书《Functional Differential Geometry》，像 SICP 和 SICM 一样用 Scheme 构建微分几何理论，对想用更熟悉的语言掌握该主题的程序员来说是极好的读物，可在出版社网站免费下载。
kachnuv_ocasek查看英文原文
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-geometry
亲历上过课仍缺乏对微分形式的直觉，反思教学方式的不足
我大学修过微分几何课，涵盖光滑流形和黎曼几何基础。虽然欣赏其理论之美，但坦白说缺乏直觉，尤其是微分形式。我明白其性质适合定义流形上的积分，但除了积分之外看不出它们还有什么用途；即便在积分中，也只是把 R^n 上的定义通过图表拉回推广到流形，这并没有解开微分形式本身为何重要的谜团。真希望当时修过大量使用向量微积分的物理课。课程确实让我成为更好的数学家，但即使能证明定理，仍感觉没有真正完全掌握这些概念。
FiberBundle查看英文原文
I took a Differential Geometry course in university. The course covered basic theory on smooth manifolds and Riemannian Geometry. While I did appreciate the beauty of the subject from a theoretical perspective, I have to admit that I somewhat lacked intuition for the material, especially with regards to differential forms. I understood that their properties make sense to define the notion of integration on a smooth manifold, but never really saw whether they had any purpose besides using them for integration. Even in their use in integration they were kind of a mystery to me, if I recall correctly we just defined integration on subset of R^n, where differential forms played the same role as d_{x_1}d_{x_2}...d_{x_n}, which is known from the riemann integral, but then to define integration on manifolds, we just used the pullback of the charts, which didn't alleviate any of the mystery and didn't add anything about why differential forms themselves are supposed to be important. I wish I had taken some physics courses, where vector calculus is used heavily and that might have helped some, but in the end I was somewhat disappointed, because even though taking the course certainly did make me a better mathematician, I still didn't have the feeling of really completely grasping the concepts, even though I could prove statements.
补充推荐两本微分形式入门经典教材
想深入学习的读者，我强烈推荐 Harold Edwards 的《Advanced Calculus: A Differential Forms Approach》和 Loring Tu 的《An Introduction to Manifolds》。前者读起来像小说，是数学阐述的佳作，还带点俏皮；后者是流形与微分形式数学入门的金标准，最简洁直接地介绍了完整理论。
bmitc查看英文原文
For anyone wanting to learn more about this, I highly recommend Advanced Calculus: A Differential Forms Approach by Harold Edwards and An Introduction to Manifolds by Loring Tu. The former reads almost like a novel and is a real treat of mathematical exposition. It's also a little quirky which is always nice. Tu's book is simply the gold standard of an introduction to the mathematics of manifolds and differential forms. It is the most concise and straightforward introduction to the full theory. It is also a wonderful book.
质疑求有直觉性讲解的入门资源，指出形式定义与直觉脱节的困境
有人能推荐微分几何和微分形式的好的入门材料吗？有没有类似《Visual Complex Analysis》那样的书？我长期好奇，但读这类文章总会在某处迷失：读完感觉似乎懂了，却对形式或流形没有任何具体直觉，尽管知道形式定义。应用例子可能有帮助，但那似乎意味着要先去学一大堆艰深的物理；或者做证明，但没有导师检查又觉得徒劳。
0xddd查看英文原文
Can anyone recommend a good introduction to differential geometry and forms? Does something analogous to "Visual Complex Analysis" exist for the topic? I have been curious to learn for a long time but, for whatever reason, always lose my way at some point with articles like this. I come away with some feeling that I understand what's going on and yet I can't say I have any concrete intuition for what a form or a manifold is despite knowing the formal definitions. I feel like applied examples would help, but at this level of math that seems to entail going on a side quest to learn a lot of difficult physics first. (Or alternatively, doing a lot of proofs, but that feels futile without having a tutor/mentor to check them.)
历次讨论
- 2020Hacker News 讨论帖145 分 · 45 评
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