经典著作都很棒：* 希尔伯特和康福森的《几何与想象》（Geometry and the imagination）* 柯朗和希尔伯特的《数学物理方法》（Methods of mathematical physics）* 斯皮瓦克的《微分几何综合教程》（A comprehensive introduction to differential geometry，以及它的同门小弟《微积分》和《流形上的微积分》）* 柯纳的《傅里叶分析》（Fourier Analysis）* 阿诺德关于……的书
The classical stuff is great: * Geometry and the imagination by Hilbert and Cohn-Vossen * Methods of mathematical physics by Courant and Hilbert * A comprehensive introduction to differential geometry by Spivak (and its little brothers Calculus and Calculus on manifolds) * Fourier Analysis by Körner * Arnold's books on…
…swim. EDIT: Also, some books by Hilbert are breathtakingly beautiful: Geometry and the Imagination (just the chapter on synthetic differential geometry is worth more than 10 other great books), and the Methods of Mathematical Physics is also great. It begins by giving three proofs of cauchy-schwartz inequality, and then goes on to give several different definitions of the eigenvectors of a matrix. Both of those make great beach…
……函数、积分方程、变分法、二次型，当然还有偏微分方程。希尔伯特本人其实和理查德·柯朗合著过一本涵盖这些主题的教科书：《数学物理方法》（Methods of Mathematical Physics）。正因如此，我觉得爱因斯坦那一代的物理学家其实很多人并不懂太多数学，单纯是因为当时根本不教。而在随后的几十年里，物理学家逐渐普遍变得……
…functions, integral equations, calculus of variations, quadratic forms and of course partial differential equations. Hilbert himself actually wrote a textbook with Richard Courant covering these topics: Methods of Mathematical Physics. As such I don't really think many physicists of Einstein's generation knew lots of mathematics simply because it wasn't taught at the time. In the decades since physicists gradually has generally become…
…applications if not in their presentation, as the interplay between pure mathematics and its applications was always an important subject to him (see also the introduction to his [and, nominally, Hilbert's] Methods of Mathematical Physics [4], the address he gave on variational methods in PDEs [5], often cited as a foundational work in finite element analysis, and, well, pretty much the entirety of What is Mathematics?[6]). [1] https://en.wikipedia.org/wiki/Nonstandard_analysis…
对，我是物理学/广义相对论（GR）背景出身，所以我看这一切都是透过 MTW 的视角。还有 Schutz 的《数学物理的几何方法》（Geometrical Methods of Mathematical Physics）。我同意我们不需要坐标，东西本身就是那个东西。但几何代数那一套总让我别扭的地方在于，他们用二重向量（bivectors）来表示面积，这感觉就不太对劲。如果你……
Yeah, I have a physics / GR background, so MTW is the lens through which I see all of this. That and Schutz's Geometrical Methods of Mathematical Physics. I agree that we don't need coordinates. Things are things. But what always got me about the geometric algebra stuff was that they used bivectors for areas, which seems like the wrong thing. If you're…
针对什么水平、哪个领域的？像《数学物理方法》（Methods of mathematical physics）这类书可能既太难又脱离你的实际需求。初学者要想用高中数学水平提升解题能力：- 波利亚的《怎样解题》（How to Solve It）- 自己挑几本数学竞赛相关的书。……
For what level and in which area? Books like _Methods of mathematical physics_ can be both too hard and irrelevant to your needs. For starters, To become a better problem solver with high-school level maths: - Polya's How to Solve It. - Books of your choice about math contests.…
经典著作都很棒：* 希尔伯特和康福森的《几何与想象》（Geometry and the imagination）* 柯朗和希尔伯特的《数学物理方法》（Methods of mathematical physics）* 斯皮瓦克的《微分几何综合教程》（A comprehensive introduction to differential geometry，以及它的同门小弟《微积分》和《流形上的微积分》）* 柯纳的《傅里叶分析》（Fourier Analysis）* 阿诺德关于……的书
The classical stuff is great: * Geometry and the imagination by Hilbert and Cohn-Vossen * Methods of mathematical physics by Courant and Hilbert * A comprehensive introduction to differential geometry by Spivak (and its little brothers Calculus and Calculus on manifolds) * Fourier Analysis by Körner * Arnold's books on…
……游泳。编辑补充：另外，希尔伯特写的一些书真是美得令人屏息：《几何与想象》（光是综合微分几何那一章，就顶得上其他 10 本好书），还有《数学物理方法》也极其精彩。开篇就给出了柯西-施瓦茨不等式的三种证明，接着又给出了矩阵特征向量的几种不同定义。这两本书都非常适合带去海滩……
…swim. EDIT: Also, some books by Hilbert are breathtakingly beautiful: Geometry and the Imagination (just the chapter on synthetic differential geometry is worth more than 10 other great books), and the Methods of Mathematical Physics is also great. It begins by giving three proofs of cauchy-schwartz inequality, and then goes on to give several different definitions of the eigenvectors of a matrix. Both of those make great beach…
……函数、积分方程、变分法、二次型，当然还有偏微分方程。希尔伯特本人其实和理查德·柯朗合著过一本涵盖这些主题的教科书：《数学物理方法》（Methods of Mathematical Physics）。正因如此，我觉得爱因斯坦那一代的物理学家其实很多人并不懂太多数学，单纯是因为当时根本不教。而在随后的几十年里，物理学家逐渐普遍变得……
…functions, integral equations, calculus of variations, quadratic forms and of course partial differential equations. Hilbert himself actually wrote a textbook with Richard Courant covering these topics: Methods of Mathematical Physics. As such I don't really think many physicists of Einstein's generation knew lots of mathematics simply because it wasn't taught at the time. In the decades since physicists gradually has generally become…
……应用（即使在表述上没体现出来），因为纯数学与其应用之间的相互交融对他而言始终是一个重要课题（另见他[以及名义上属于希尔伯特]的《数学物理方法》[4]的前言，他在偏微分方程变分法方面的演讲[5]——常被誉为有限元分析的奠基之作，还有，基本上可以说整本《什么是数学？》[6]）。[1] https://en.wikipedia.org/wiki/Nonstandard_analysis…
…applications if not in their presentation, as the interplay between pure mathematics and its applications was always an important subject to him (see also the introduction to his [and, nominally, Hilbert's] Methods of Mathematical Physics [4], the address he gave on variational methods in PDEs [5], often cited as a foundational work in finite element analysis, and, well, pretty much the entirety of What is Mathematics?[6]). [1] https://en.wikipedia.org/wiki/Nonstandard_analysis…
对，我是物理学/广义相对论（GR）背景出身，所以我看这一切都是透过 MTW 的视角。还有 Schutz 的《数学物理的几何方法》（Geometrical Methods of Mathematical Physics）。我同意我们不需要坐标，东西本身就是那个东西。但几何代数那一套总让我别扭的地方在于，他们用二重向量（bivectors）来表示面积，这感觉就不太对劲。如果你……
Yeah, I have a physics / GR background, so MTW is the lens through which I see all of this. That and Schutz's Geometrical Methods of Mathematical Physics. I agree that we don't need coordinates. Things are things. But what always got me about the geometric algebra stuff was that they used bivectors for areas, which seems like the wrong thing. If you're…
针对什么水平、哪个领域的？像《数学物理方法》（Methods of mathematical physics）这类书可能既太难又脱离你的实际需求。初学者要想用高中数学水平提升解题能力：- 波利亚的《怎样解题》（How to Solve It）- 自己挑几本数学竞赛相关的书。……
For what level and in which area? Books like _Methods of mathematical physics_ can be both too hard and irrelevant to your needs. For starters, To become a better problem solver with high-school level maths: - Polya's How to Solve It. - Books of your choice about math contests.…