出版社: Cambridge University Press
副标题: But Need to Know for Graduate School
出版年: 20011112
页数: 376
定价: USD 44.99
装帧: Paperback
ISBN: 9780521797078
内容简介 · · · · · ·
Beginning graduate students in mathematics and other quantitative subjects are expected to have a daunting breadth of mathematical knowledge. But few have such a background. This 2002 book will help students to see the broad outline of mathematics and to fill in the gaps in their knowledge. The author explains the basic points and a few key results of all the most important und...
Beginning graduate students in mathematics and other quantitative subjects are expected to have a daunting breadth of mathematical knowledge. But few have such a background. This 2002 book will help students to see the broad outline of mathematics and to fill in the gaps in their knowledge. The author explains the basic points and a few key results of all the most important undergraduate topics in mathematics, emphasizing the intuitions behind the subject. The topics include linear algebra, vector calculus, differential geometry, real analysis, pointset topology, probability, complex analysis, abstract algebra, and more. An annotated bibliography then offers a guide to further reading and to more rigorous foundations. This book will be an essential resource for advanced undergraduate and beginning graduate students in mathematics, the physical sciences, engineering, computer science, statistics, and economics who need to quickly learn some serious mathematics.
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读书笔记 · · · · · ·
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It covers some basic knowledge of point set topology, including definition of topology space, open and closed set, induced topology, Hausdorff space, base, second countable space, compactness, connectedness, continuity... Almost all the results are ordinary, except the last section: Zariski topology of commutative rings. I recommend to skip it if not having a basic knowledge of commutative ring t...
20150710 16:21
It covers some basic knowledge of point set topology, including definition of topology space, open and closed set, induced topology, Hausdorff space, base, second countable space, compactness, connectedness, continuity...Almost all the results are ordinary, except the last section: Zariski topology of commutative rings. I recommend to skip it if not having a basic knowledge of commutative ring theory.Besides, I don't think listing all the definition and theorem then proving them one by one is a good idea to learn mathematics, except that you want a pure proof exercise (but I think it is a wast of time).回应 20150710 16:21 
You need to know: 1. Structure of /公式内容已省略/. 2. Differential of a function is a linear mapping, so Jacobi matrix replaces the role of derivative /公式内容已省略/. Actually, to define derivative, linear mapping /公式内容已省略/ is a better idea than a number /公式内..
20150705 20:28
You need to know:1. Structure of ($R^n$).2. Differential of a function is a linear mapping, so Jacobi matrix replaces the role of derivative ($\frac{dy}{dx}$). Actually, to define derivative, linear mapping ($\frac{dy}{dx}:R^n \to R^m$) is a better idea than a number ($f'(x)$).3. Inverse function theorem and implicit function theorem.Books:Spivak's Calculus on Manifold is great. I prefer Zurich's masterpiece.回应 20150705 20:28 
PS: \epsilon = /公式内容已省略/ and \delta = /公式内容已省略/. Some basic knowledge of one variable calculus. Actually it is boring. Limit, continuity, differentiation, integration, point wise and uniform convergence of function series. Uniform convergence can be understand better after studying metric space (it is just limit of f...
20150705 20:21
PS: \epsilon = ($\epsilon$) and \delta = ($\delta$).Some basic knowledge of one variable calculus. Actually it is boring.Limit, continuity, differentiation, integration, point wise and uniform convergence of function series. Uniform convergence can be understand better after studying metric space (it is just limit of function series in a function space with uniform metric).Many people speak highly of Michael Spivak's Calculus. And I recommend V. A. Zurich's Mathematical Analysis.回应 20150705 20:21 
I have learned some linear algebra, so the first chapter is not very interesting. For freshmen, I suggest you to pay special attention to the following points: 0. Definition of vector space. 1. The relation between linear mapping and matrix (Matrix is just a representation of linear mapping and different matrices representation of one linear mapping is similar matrix). 2. Definition and basic ...
20150705 20:13
I have learned some linear algebra, so the first chapter is not very interesting.For freshmen, I suggest you to pay special attention to the following points:0. Definition of vector space.1. The relation between linear mapping and matrix (Matrix is just a representation of linear mapping and different matrices representation of one linear mapping is similar matrix).2. Definition and basic properties of eigenvalues and eigenvectors.3.* Dual space. Many textbook neglect this important topic, which appears frequently in analysis(differential form), geometry(covector) and etc..FYI, I recommend Sheldon, Axler, Linear Algebra Done Right 3ed.回应 20150705 20:13

I have learned some linear algebra, so the first chapter is not very interesting. For freshmen, I suggest you to pay special attention to the following points: 0. Definition of vector space. 1. The relation between linear mapping and matrix (Matrix is just a representation of linear mapping and different matrices representation of one linear mapping is similar matrix). 2. Definition and basic ...
20150705 20:13
I have learned some linear algebra, so the first chapter is not very interesting.For freshmen, I suggest you to pay special attention to the following points:0. Definition of vector space.1. The relation between linear mapping and matrix (Matrix is just a representation of linear mapping and different matrices representation of one linear mapping is similar matrix).2. Definition and basic properties of eigenvalues and eigenvectors.3.* Dual space. Many textbook neglect this important topic, which appears frequently in analysis(differential form), geometry(covector) and etc..FYI, I recommend Sheldon, Axler, Linear Algebra Done Right 3ed.回应 20150705 20:13 
PS: \epsilon = /公式内容已省略/ and \delta = /公式内容已省略/. Some basic knowledge of one variable calculus. Actually it is boring. Limit, continuity, differentiation, integration, point wise and uniform convergence of function series. Uniform convergence can be understand better after studying metric space (it is just limit of f...
20150705 20:21
PS: \epsilon = ($\epsilon$) and \delta = ($\delta$).Some basic knowledge of one variable calculus. Actually it is boring.Limit, continuity, differentiation, integration, point wise and uniform convergence of function series. Uniform convergence can be understand better after studying metric space (it is just limit of function series in a function space with uniform metric).Many people speak highly of Michael Spivak's Calculus. And I recommend V. A. Zurich's Mathematical Analysis.回应 20150705 20:21 
You need to know: 1. Structure of /公式内容已省略/. 2. Differential of a function is a linear mapping, so Jacobi matrix replaces the role of derivative /公式内容已省略/. Actually, to define derivative, linear mapping /公式内容已省略/ is a better idea than a number /公式内..
20150705 20:28
You need to know:1. Structure of ($R^n$).2. Differential of a function is a linear mapping, so Jacobi matrix replaces the role of derivative ($\frac{dy}{dx}$). Actually, to define derivative, linear mapping ($\frac{dy}{dx}:R^n \to R^m$) is a better idea than a number ($f'(x)$).3. Inverse function theorem and implicit function theorem.Books:Spivak's Calculus on Manifold is great. I prefer Zurich's masterpiece.回应 20150705 20:28 
It covers some basic knowledge of point set topology, including definition of topology space, open and closed set, induced topology, Hausdorff space, base, second countable space, compactness, connectedness, continuity... Almost all the results are ordinary, except the last section: Zariski topology of commutative rings. I recommend to skip it if not having a basic knowledge of commutative ring t...
20150710 16:21
It covers some basic knowledge of point set topology, including definition of topology space, open and closed set, induced topology, Hausdorff space, base, second countable space, compactness, connectedness, continuity...Almost all the results are ordinary, except the last section: Zariski topology of commutative rings. I recommend to skip it if not having a basic knowledge of commutative ring theory.Besides, I don't think listing all the definition and theorem then proving them one by one is a good idea to learn mathematics, except that you want a pure proof exercise (but I think it is a wast of time).回应 20150710 16:21

It covers some basic knowledge of point set topology, including definition of topology space, open and closed set, induced topology, Hausdorff space, base, second countable space, compactness, connectedness, continuity... Almost all the results are ordinary, except the last section: Zariski topology of commutative rings. I recommend to skip it if not having a basic knowledge of commutative ring t...
20150710 16:21
It covers some basic knowledge of point set topology, including definition of topology space, open and closed set, induced topology, Hausdorff space, base, second countable space, compactness, connectedness, continuity...Almost all the results are ordinary, except the last section: Zariski topology of commutative rings. I recommend to skip it if not having a basic knowledge of commutative ring theory.Besides, I don't think listing all the definition and theorem then proving them one by one is a good idea to learn mathematics, except that you want a pure proof exercise (but I think it is a wast of time).回应 20150710 16:21 
You need to know: 1. Structure of /公式内容已省略/. 2. Differential of a function is a linear mapping, so Jacobi matrix replaces the role of derivative /公式内容已省略/. Actually, to define derivative, linear mapping /公式内容已省略/ is a better idea than a number /公式内..
20150705 20:28
You need to know:1. Structure of ($R^n$).2. Differential of a function is a linear mapping, so Jacobi matrix replaces the role of derivative ($\frac{dy}{dx}$). Actually, to define derivative, linear mapping ($\frac{dy}{dx}:R^n \to R^m$) is a better idea than a number ($f'(x)$).3. Inverse function theorem and implicit function theorem.Books:Spivak's Calculus on Manifold is great. I prefer Zurich's masterpiece.回应 20150705 20:28 
PS: \epsilon = /公式内容已省略/ and \delta = /公式内容已省略/. Some basic knowledge of one variable calculus. Actually it is boring. Limit, continuity, differentiation, integration, point wise and uniform convergence of function series. Uniform convergence can be understand better after studying metric space (it is just limit of f...
20150705 20:21
PS: \epsilon = ($\epsilon$) and \delta = ($\delta$).Some basic knowledge of one variable calculus. Actually it is boring.Limit, continuity, differentiation, integration, point wise and uniform convergence of function series. Uniform convergence can be understand better after studying metric space (it is just limit of function series in a function space with uniform metric).Many people speak highly of Michael Spivak's Calculus. And I recommend V. A. Zurich's Mathematical Analysis.回应 20150705 20:21 
I have learned some linear algebra, so the first chapter is not very interesting. For freshmen, I suggest you to pay special attention to the following points: 0. Definition of vector space. 1. The relation between linear mapping and matrix (Matrix is just a representation of linear mapping and different matrices representation of one linear mapping is similar matrix). 2. Definition and basic ...
20150705 20:13
I have learned some linear algebra, so the first chapter is not very interesting.For freshmen, I suggest you to pay special attention to the following points:0. Definition of vector space.1. The relation between linear mapping and matrix (Matrix is just a representation of linear mapping and different matrices representation of one linear mapping is similar matrix).2. Definition and basic properties of eigenvalues and eigenvectors.3.* Dual space. Many textbook neglect this important topic, which appears frequently in analysis(differential form), geometry(covector) and etc..FYI, I recommend Sheldon, Axler, Linear Algebra Done Right 3ed.回应 20150705 20:13
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订阅关于All the Mathematics You Missed的评论:
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0 有用 Girasol 20190122
一本试图以数学打通各个理工科的自学教辅/休闲读物，将近400页居然能覆盖如此广泛的数学~做到每一专题都是brief overview实属不易，值得赞赏的是很多原创的思维隐藏在了初次证明里面，然并，原来懂的还是一眼就明白了，原来读不懂的还是不懂...
5 有用 安雪仁次 20140429
终于翻完了这本小巨著，深觉会的部分讲得好浅，不会的部分看这个也不能入门QAQ
1 有用 恆轉如瀑流 20180407
假装看完了。。。
0 有用 感冒的冬天 20180213
以前读的，实际每个章节讲得超级简单，也就给出了基本定义。作为中学生科普或许比较合适。
0 有用 头上有角 20150818
理工科读phd前暑假必读书籍之一
0 有用 Girasol 20190122
一本试图以数学打通各个理工科的自学教辅/休闲读物，将近400页居然能覆盖如此广泛的数学~做到每一专题都是brief overview实属不易，值得赞赏的是很多原创的思维隐藏在了初次证明里面，然并，原来懂的还是一眼就明白了，原来读不懂的还是不懂...
0 有用 huyan00 20181228
大概高中生就能读，没有看最后几章，感觉最难的是微分形式那章偷懒放下了，对本科数学各部分的核心做了介绍，想了解数学各个领域都是研究什么可以从这本书开始
1 有用 恆轉如瀑流 20180407
假装看完了。。。
0 有用 感冒的冬天 20180213
以前读的，实际每个章节讲得超级简单，也就给出了基本定义。作为中学生科普或许比较合适。
0 有用 Harrison 20160824
查漏补缺...可怕的PhD life已经来了