多元函数
Functions of Several Variables
Wendell Fleming、W. H. Fleming · 初版 1963 年
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新版FUNCTIONS OF SEVERAL VARIABLES经过了全面大幅的修订。与第一版相同，本书系统详尽地介绍了微分与积分学，包括流形上微分形式的积分。新增的初等拓扑学章节使本书作为高等微积分教材更趋完备；此外还增补了介绍在热力学、流体力学和经典刚体力学中物理应用的相关章节。
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……好得多。第二遍读得很慢、很仔细，我细细咀嚼那些证明，试图理解它们为什么能成立，花时间去建立直觉等等。接着我又仔细研读了 Fleming 的《Functions of Several Variables》，并在我的职业生涯中研究了各种各样相关的课题。后来在我的博士分析学资格考试中，对 Rudin 和 Fleming 的仔细研读让我拿到了系里的最高分。教训/心得是：那种……
…much better. That second pass was slow, careful, where I chewed on the proofs to try to understand why they worked, took the time to develop intuition, etc. Then I took a careful pass through Fleming, Functions of Several Variables and, in my career, studied a wide variety of related topics. Then on my Ph.D. analysis qualifying exam, those passes through Rudin and Fleming got me the best score in the department. Lesson: That…
……乐在其中甚至顺利读完。提示：找个好老师！然后进阶的话，比较流行的是 Spivak 的《Calculus on Manifolds》，主要因为它很薄。其实，它太薄了。我更喜欢 Fleming 的《Functions of Several Variables》，直到读到外代数那一章；到那时，如果你感兴趣，可以去买 Henri Cartan 那本薄薄的《Differential Forms》（ISBN 0-486-45010-4, Dover, Mineola, NY, 2006 英文版）。在这之间……
…enjoy it or even get through it. Hint: Get a really good teacher! Then for more, popular is Spivak, 'Calculus on Manifolds', mostly because it is short. Actually, it's too short. I prefer Fleming, 'Functions of Several Variables' until get to the exterior algebra chapter at which time, if care, can now get the thin Henri Cartan, {\it Differential Forms,\/} ISBN 0-486-45010-4, Dover, Mineola, NY, 2006.\ \ in English. Between…
……Rao 的《Linear Statistical Inference and Its Applications: Second Edition》（ISBN 0-471-70823-2, John Wiley and Sons, New York, 1967）。凸性（convexity）方面一个很好的入门书是 Wendell H. Fleming 的《Functions of Several Variables》（Addison-Wesley, Reading, Massachusetts, 1965）。凸性里绝对好玩、堪比甜点冰淇淋的是琴生不等式（Jensen's inequality）；立刻就能用它来证明一大堆经典不等式。天呐，看看……
…Rao, {\it Linear Statistical Inference and Its Applications:\ \ Second Edition,\/} ISBN 0-471-70823-2, John Wiley and Sons, New York, 1967.\ \ A good start on convexity is Wendell H.\ Fleming, {\it Functions of Several Variables,\/} Addison-Wesley, Reading, Massachusetts, 1965.\ \ Total, fun, dessert ice cream on convexity is Jensen's inequality; right away can use it to prove a lot of classic inequalities. Gee, look on the…
……从一些最好的数学经典教材中自学了大量内容，比如我把 Rudin 的《Principles》刷了第二遍，仔细啃完了 Halmos 的《Finite Dimensional Vector Spaces》、Fleming 的《Functions of Several Variables》、von Neumann 的《Quantum Mechanics》中的数学部分，还有很多很多，整整一大堆。当我去读博士时，入学前学到的东西已经几乎足够应付课程要求了。至于研究方面，……
…a lot of independent learning from some of the best math texts, e.g., I took a second pass through Rudin's 'Principles', went carefully through Halmos's 'Finite Dimensional Vector Spaces', Fleming's 'Functions of Several Variables', the math parts of von Neumann's 'Quantum Mechanics', and much more, a big stack more. When I went for my Ph.D., what I had learned before I entered was nearly enough for the course work. For the research,…
……关于线性代数中仅靠高斯消元法就能轻松看到的线性变换性质的局部非线性推广。想要参考来源？W. Fleming 的《Functions of Several Variables》里有很好的阐述。如果我没记错的话，书里还有一个基于压缩映射（contractive mappings）的绝妙证明。这样一来，你在连续性（continuity）和线性（linearity）方面就有了一个很好的起点。
…local, non-linear generalization of what you will see really easily for linear transformations in linear algebra via, right, just Gauss elimination. For a source? There is a good treatment in W. Fleming, Functions of Several Variables. IIRC, there is also a cute proof based on contractive mappings. So, by then you will have a good start on both continuity and linearity.
……在课程中看清楚具体讲了什么、怎么推导的以及为什么；(iii) 同时也学习或浏览一两本其他旗鼓相当或同类型的书，比如 Spivak 的《Calculus on Manifolds》、Fleming 的《Functions of Several Variables》、Kolmogorov and Fomin、Apostol 等等；(iv) 把 Baby Rudin 再刷一遍。总之，Baby Rudin 让很多学生望而却步，打击了太多人。对许多学生来说，Baby Rudin 看起来难得令人望而生畏……
…here to see during the course some clarity in what is going on, how, and why; (iii) also study or glance at one or two other competitive or similar books, e.g., Spivak, Calculus on Manifolds, Fleming, Functions of Several Variables, Kolmogorov and Fomin, Apostol, and more, and (iv) take a second pass through Baby Rudin. Net, Baby Rudin has discouraged a lot of students, too many. To many students, Baby Rudin looks forbiddingly…