…solving, contests, and inequalities, say by Polya or Andreescu. ... a higher-level view of day-to-day mathematical practice. There probably isn't one book for this, but I'd recommend Loomis and Sternberg's Advanced Calculus as a summation of linear algebra and calculus on manifolds; you'd also need to read on complex and functional analysis, algebra (Lang and the unofficial companion volume?), topology ... ... a broad but…
……Company, Inc., Princeton, New Jersey。Walter Rudin，《Principles of Mathematical Analysis, Third Edition》，McGraw-Hill。Michael Spivak，《Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus》，W. A. Benjamin, New York。成功啃完这三本，就足够理解基于证明的大学数学了！这三本都很老；尤其是 Halmos 写了第一版……
…Company, Inc., Princeton, New Jersey. Walter Rudin, 'Principles of Mathematical Analysis, Third Edition', McGraw-Hill. Michael Spivak, 'Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus', W. A. Benjamin, New York. Working successfully through those three is quite sufficient to understand proof-based college math! Those three are all old; in particular Halmos wrote the first edition…
……解题、竞赛和不等式，比如 Polya 或 Andreescu 的。……对日常数学实践更高层次的视角。可能没有哪一本书能涵盖这个，但我推荐 Loomis 和 Sternberg 的《Advanced Calculus》，作为线性代数和流形上微积分的总结；你还需要读复分析和泛函分析、代数（Lang 和非官方配套卷？）、拓扑学……广泛但……
…solving, contests, and inequalities, say by Polya or Andreescu. ... a higher-level view of day-to-day mathematical practice. There probably isn't one book for this, but I'd recommend Loomis and Sternberg's Advanced Calculus as a summation of linear algebra and calculus on manifolds; you'd also need to read on complex and functional analysis, algebra (Lang and the unofficial companion volume?), topology ... ... a broad but…
……Company, Inc., Princeton, New Jersey。Walter Rudin，《Principles of Mathematical Analysis, Third Edition》，McGraw-Hill。Michael Spivak，《Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus》，W. A. Benjamin, New York。成功啃完这三本，就足够理解基于证明的大学数学了！这三本都很老；尤其是 Halmos 写了第一版……
…Company, Inc., Princeton, New Jersey. Walter Rudin, 'Principles of Mathematical Analysis, Third Edition', McGraw-Hill. Michael Spivak, 'Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus', W. A. Benjamin, New York. Working successfully through those three is quite sufficient to understand proof-based college math! Those three are all old; in particular Halmos wrote the first edition…