出版日期:2016-1-1
ISBN:9787519200191
作者:Rodney Coleman
页数:249页
内容概要
Rodney Coleman（R.科尔曼,法国）是国际知名学者，在数学界享有盛誉。本书凝聚了作者多年科研和教学成果，适用于科研工作者、高校教师和研究生。
书籍目录
Preface
1 Normed Vector Spaces
1.1 First Notions
1.2 Limits and Continuity
1.3 Open and Closed Sets
1.4 Compactness
1.5 Banach Spaces
1.6 Linear and Polynomial Mappings
1.7 Normed Algebras
1.8 The Exponential Mapping
Appendix： The Fundamental Theorem of Algebra
2 Differentiation
2.1 Directional Derivatives
2.2 The Differential
2.3 Differentials of Compositions
2.4 Mappings of Class C1
2.5 Extrema
2.6 Differentiability of the Norm
Appendix： Gateaux Differentiability
3 Mean Value Theorems
3.1 Generalizing the Mean Value Theorem
3.2 Partial Differentials
3.3 Integration
3.4 Differentiation under the Integral Sign
4 Higher Derivatives and Differentials
4.1 Schwarz's Theorem
4.2 Operationson Ck—Mappings
4.3 Multilinear Mappings
4.4 Higher Differentials
4.5 Higher Differentials and Higher Derivatives
4.6 Cartesian Product Image Spaces
4.7 Higher Partial Differentials
4.8 Generalizing Ck to Normed Vector Spaces
4.9 Leibniz's Rule
5 Taylor Theorems and Applicahons
5.1 Taylor Formulas
5.2 Asymptotic Developments
5.3 Extrema： Second—Order Conditions
Appendix： Homogeneous Polynomials
6 Hilbert Spaces
6.1 Basic Notions
6.2 Projections
6.3 The Distance Mapping
6.4 The Riesz Representation Theorem
7 Convex Functions
7.1 Preliminary Results
7.2 Continuity of Convex Functions
7.3 Differentiable Convex Functions
7.4 Extrema of Convex Functions
Appendix： Convex Polyhedra
8 The Inverse and Implicit Mapping Theorems
8.1 The Inverse Mapping Theorem
8.2 The Implicit Mapping Theorem
8.3 The Rank Theorem
8.4 Constrained Extrema
Appendix 1： Bijective Continuous Linear Mappings
Appendix 2： Contractions
9 Vector Fields
9.1 Existence of Integral Curves
9.2 Initial Conditions
9.3 Geometrical Properties of Integral Curves
9.4 Complete Vector Fields
Appendix： A Useful Result on Smooth Functions
10 The Flow of a Vector Field
10.1 Continuity of the Flow
10.2 Differentiability of the Flow
10.3 Higher Differentiability of the Flow
10.4 The Reduced Flow
10.5 One—Parameter Subgroups
11 The Calculus of Variations： An Introduction
11.1 The Space C1（I， E）
11.2 Lagrangian Mappings
11.3 Fixed Endpoint Problems
11.4 Euler—LagrangeEquations
11.5 Convexity
11.6 The Class of an Extremal
References
Index
作者简介
本书适合高年级本科生或低年级研究生学习赋范向量空间上的微积分。书中不仅有成熟的数学模型，还有基础的微积分和线性代数。在必要处对重要拓扑学和泛函分析也作了介绍。
为了讲述赋范向量空间上的微积分在多变量函数基础微积分上的应用，该书是为数不多的几本能够连接初级文本和高级文本的教科书。书中穿插的该理论非平凡解的应用以及有趣的练习为读者学习赋范向量空间上的微积分提供了动力。目次：赋范向量空间；微分法；中值定理；高阶倒数与微分；泰勒定理及应用；Hilbert空间；凸函数；逆映射定理和隐式映射定理；向量场；向量场流；变分泛函：导论；参考文献；索引。
读者对象：数学专业高年级本科生及低年级研究生以及相关专业的科研工作者、技术人员。