流形上的微积分
Calculus on manifolds
Michael Spivak · 初版 1965 年
22人推荐
48位读者提及
65次提及
为什么值得读
- 行文简练，远超平庸的普通本科教材
- 采用函数记号，反思并纠正古典记号的混淆
- 运用外代数清晰证明n维广义斯托克斯定理
一本供本科生 calculus of variations 课程使用的补充教材，介绍了该领域基于 measure theoretic geometry 的现代方法。书中通过对 Plateau's problem 的深入剖析，呈现了 Varifold geometry。
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……对于建立直觉很有用（Landau & Lifshitz 的《Statistical Physics》就不错）。接下来应该学经典场论，也就是电磁学（J.D. Jackson）。然后，我建议学学《Calculus on Manifolds》（Spivak 著）和微分几何（我现在一时想不起这方面的好教材了）。如果你学到了这一步，你对理解 GR 和 QFT 所需的数学知识就已经有了扎实的掌握。Zee 有一本……
…useful for building up an intuition (Landau & Lifshitz - Statistical Physics is good). Classical Field Theory, i.e. Electromagnetism should now be learned (J.D. Jackson). Then, I'd recommend learning Calculus on Manifolds (Spivak) and Differential Geometry (I can't remember a good textbook on this right now). If you're at this point you will now have a solid understanding of the maths needed to know GR and QFT. Zee has…
……行列式……我刚翻看了（在我这本的第 99 页），他刚给出定义就承认这是个“有点拐弯抹角的做法”！还有其他参考书看起来干净利落得多（比如 Spivak 的《Calculus on Manifolds》，用的是外代数）。
…determinant... I have just read it (page 99 of my copy) and he admits that it is a "somewhat roundabout procedure", just after giving the definition! There's other references that seem much cleaner (e.g. Spivak's calculus on manifolds, using exterior algebra).
……df(x)/dx 会让数学、尤其是微积分入门对初学者来说变得极其令人困惑；他极力主张数学教育工作者使用更规范的现代记号。2 在他那本精彩的《Calculus on Manifolds》[40] 中，Michael Spivak 使用了函数式记号。在第 44 页，他探讨了经典记号存在的一些问题。我们摘录了一段尤为精彩的内容：用经典记号单纯表述[链式法则]……
…df(x)/dx makes mathematics, and especially introductory calculus, extremely confusing for beginning students; and he enjoins mathematics educators to use more formal modern notation. 2 In his beautiful book Calculus on Manifolds [40], Michael Spivak uses functional notation. On p. 44 he discusses some of the problems with classical notation. We excerpt a particularly juicy passage: The mere statement of [the chain rule] in classical…
在你们学科可能确实如此，但在本科数学里有一些非常出色、简练的教材。比如 Apostol 的《Calculus》或者 Spivak 的《Calculus on Manifolds》。虽然绝对谈不上容易读懂，但也绝不是 Pearson 或 Wiley 那种到处注水的口水书。
That may well be true in your discipline, but there are some excellent, concise texts in undergraduate mathematics. For example, Apostol's Calculus or Spivak's Calculus on Manifolds. Not the most accessible texts by a longshot, but not your average Pearson or Wiley drivel either.
外积/wedge product 实际上和 cross product（叉乘）关系非常紧密，它的作用是把叉乘推广到 n 维。你可以读读 Spivak 或 Munkres 的书，分别是《Calculus on Manifolds》和《Analysis on Manifolds》。它们的主要用途之一是证明 n 维下的推广 Stokes 定理。3D rotor 和四元数同构，所以尽管去改你的变量名吧……
The exterior/wedge product is actually very closely related to the cross product, and works to generalize the cross product to n dimensions. You can read Spivak or Munkres, "Calculus on Manifolds" and "Analysis on Manifolds" respectively. One of their main uses is to prove the generalized Stokes theorem in n dimensions. 3D rotors are isomorphic to quaternions, so go ahead and rename your variables…
在 Reddit 和其他地方确实有这么一种现象：有人发帖问“我微积分课学得很吃力，有什么好书推荐吗？”，结果真有人一本正经地回答《Calculus on Manifolds》。说实话，CoM 确实是经典，是一本真正的大作，但它只对达到了一定 mathematical maturity（数学成熟度）的人有用。而提问者显然大概率还没到那个程度。这种现象的另一个翻版是……
For sure there is an effect on Reddit and other places where someone will post a question such as, "I'm having trouble with my Calc class, what is a good book?" and people seriously answer Calculus on Manifolds. Now, CoM is a classic, a real great book, but it is useful only to people who have reached a certain level of mathematical maturity. That, presumably, is not the questioner. A version of this is that…