Preface
Chapter 1Set，Structure of Operation on Set
1.1Sets，the Relations and Operations between Sets
1.1.1Relations between sets
1.1.2Operations between sets
1.1.3Mappings between sets
1.2Structures of Operations on Sets
1.2.1Groups，rings，fields，and linear spaces
1.2.2Group theory，some important groups
1.2.3Subgroups，product groups，quotient groups
Exercise 1
Chapter 2Linear Spaces and Linear Transformations
2.1Linear Spaces
2.1.1Examples
2.1.2Bases of linear spaces
2.1.3Subspaces and product/directsum/quitient spaces
2.1.4Inner product spaces
2.1.5Dual spaces
2.1.6Structures of linear spaces
2.2Linear Transformations
2.2.1Linear operator spaces
2.2.2Conjugate operators of linear operators
2.2.3Multilinear algebra
Exercise 2
Chapter 3Basic Knowledge of Point Set Topology
3.1Metric Spaces，Normed Linear Spaces
3.1.1Metric spaces
3.1.2Normed linear spaces
3.2Topological Spaces
3.2.1Some definitions in topological spaces
3.2.2Classification of topological spaces
3.3Continuous Mappings on Topological Spaces
3.3.1Mappings between topological spaces，continuity of mappings
3.3.2Subspaces，product spaces，quotient spaces
3.4Important Properties of Topological Spaces
3.4.1Separation axioms of topological spaces
3.4.2Connectivity of topological spaces
3.4.3Compactness of topological spaces
3.4.4Topological linear spaces
Exercise 3
Chapter 4Foundation of Functional Analysis
4.1Metric Spaces
4.1.1Completion of metric spaces
4.1.2Compactness in metric spaces
4.1.3Bases of Banach spaces
4.1.4Orthgoonal systems in Hilbert spaces
4.2Operator Theory
4.2.1Linear operators on Banach spaces
4.2.2Spectrum theory of bounded linear operators
4.3Linear Functional Theory
4.3.1Bounded linear functionals on normed linear spaces
4.3.2Bounded linear functionals on Hilbert spaces
Exercise 4
Chapter 5Distribution Theory
5.1Schwartz Space，Schwartz Distribution Space
5.1.1Schwartz space
5.1.2Schwartz distribution space
5.1.3Spaces ERn，DRn and their distribution spaces
5.2Fourier Transform on LpRn，1≤p≤2
5.2.1Fourier transformations on L1Rn
5.2.2Fourier transformations on L2Rn
5.2.3Fourier transformations on LpRn，15.3Fourier Transform on Schwartz Distribution Space
5.3.1Fourier transformations of Schwartz functions
5.3.2Fourier transformations of Schwartz distributions
5.3.3Schwartz distributions with compact supports
5.3.4Fourier transformations of convolutions of Schwartz distributions
5.4Wavelet Analysis
5.4.1Introduction
5.4.2Continuous wavelet transformations
5.4.3Discrete wavelet transformations
5.4.4Applications of wavelet transformations
Exercise 5
Chapter 6Calculus on Manifolds
6.1Basic Concepts
6.1.1Structures of differential manifolds
6.1.2Cotangent spaces，tangent spaces
6.1.3Submanifolds
6.2External Algebra
6.2.1(r，s)type tensors，(r，s)type tensor spaces
6.2.2Tensor algebra
6.2.3Grassmann algebra (exterior algebra)
6.3Exterior Differentiation of Exterior Differential Forms
6.3.1Tensor bundles and vector bundles
6.3.2Exterior differentiations of exterior differential form
6.4Integration of Exterior Differential Forms
6.4.1Directions of smooth manifolds
6.4.2Integrations of exterior differential forms on directed manifold M
6.4.3Stokes formula
6.5Riemann Manifolds， Mathematics and Modern Physics
6.5.1Riemann manifolds
6.5.2Connections
6.5.3Lie group and movingframe method
6.5.4Mathematics and modern physics
Exercise 6
Chapter 7Complimentary Knowledge
7.1Variational Calculus
7.1.1Variation and variation problems
7.1.2Variation principle
7.1.3More general variation problems
7.2Some Important Theorems in Banach Spaces
7.2.1StoneWeierstrass theorems
7.2.2Implicit and inversemapping theorems
7.2.3Fixed point theorems
7.3Haar Integrals on Locally Compact Groups
Exercise 7
References
Index
內容試閱：
The new century，the 21st century，has come.It indicates that the rapid development of science and technology，as well as productive activities require a new level of ability of scientists and technicians.It not only asks for profound knowledge and practical ability，but also for lofty ideals，excellent morality，and exquisite thinking.Especially，the great new era asks for extensive modern knowledge，smart and creative thoughts，sensitive and flexible using ability of mathematics from those members who are working in numerous areas of natural science.Clearly，the traditional course advanced calculus could not meet these new requirements.
On the other hand，various excellent concepts，valuable theories，powerful methods in modern applied mathematics are permeating into lots of scientific fields deep.From abstract theory to reality objects，from top to base，modern mathematics and natural sciences are united closely，indeed.Thus，a course of modern applied mathematics must be presented in universities after that of advanced calculus. Hence，it is extremely urgent nowadays to write a textbook of foundations of modern applied mathematics.
The contents of modern applied mathematics are very wide，the theory of it is very deep，and the knowledge included is very powerful.We mainly aim to prepare the basic knowledge and technical abilities of modern applied mathematics for undergraduate students who major in natural sciences，such as physics，astronomy，computer science，chemistry，geology，geography，biology，as well as life science.Thus，at the beginning of the 1990s，a course of modern applied mathematics was set after that of the advanced mathematics for Kuang Yaming Honors School of Nanjing University.It took one semester，five class hours per week.Its contents were mainly Lebesgue integral and differential geometry.Its results were beneficial，indeed.Then，we arranged and moved the content of Lebesgue integral into the textbook Advanced Mathematics published in 2003，and began to write Foundations of Modern Applied Mathematics. Before accomplishing the manuscript，we have printed lecture notes twice in 2007 and 2009，and used them as teaching material for 10 years in Kuang Yaming Honors School of Nanjing University and modified them continuously.
The contents of this book are arranged as follows：
Basic knowledge of set theory and modern algebra，introduced in Chapter 1.The first part of this chapter is set theory，including concepts，operations and important properties of sets and mappings between sets.The second part is modern algebra，a main branch of modern mathematics，in which the operation structure of sets is described and studied.It contains several main structures of sets，such as groups，fields，and linear spaces，and particularly how to generate certain new groups from a given group，such as subgroup，product group，and quotient group.Then，several useful groups，such as transformation groups，permutation groups，circulate groups； and their properties are concerned.
Linear spaces and linear transformations，set out in Chapter 2. These are the continuing contents of linear algebra in advanced mathematics，and also are the parts of modern linear algebra involving the structures of orthogonal geometry and skew geometry which play important roles in modern mathematics and modern physics. In the point of view of operation structures in sets，we guide our students to recognize the significance and importance of linear spaces and linear transformations based on a higher level of spaces and transforms.Finally，the theory of multilinear algebra is presented as the essential knowledge of tensor theory and differential geometry.
Point set topology，a corner stone of modern mathematics，is presented in Chapter 3.Both topological and operation structures of sets are essential and intrinsic.A variety of deeper properties of sets can be described by topological structures. A set endowed topological structure is called topological space，this concept comes from reality and Euclidean space.It is highly abstract，and implies both abundant ideas and dedicated methods of modern mathematics. We suggest our students to understand abstract definitions with visual(形象化的、栩栩如生的) examples.Specifically，we emphasize how to generate some new topological spaces from given spaces，such as subtopological space，product topological space，and quotient topological space； what are the topological structures of these new spaces? The other two concepts are very significant and very useful： a continuous mapping between spaces and the compactness of topological spaces.The classification，separability，as well as connectivity of topological spaces are presented at the end of this chapter.
Functional analysis，an indispensable part of modern applied mathematics，presented in Chapter 4.Its contents are metric space theory，linear operator theory，and linear functional theory.Firstly，about metric space theory，we prove the completion theorem and show certain useful properties of complete metric spaces. Furthermore，various kinds of compactness in metric spaces，such as the compactness，countable compactness，sequential compactness，accumulative compactness and local compactness are defined exactly.Then，two criteria of sequential compactness in normed linear spaces are listed.The Schauder base of Banach spaces，as a generalized concept of finite base of the linear space in Chapter 2，and the orthogonal expansion in Hilbert spaces，as a generalized method of Fourier series in advanced mathematics，both are presented clearly.Secondly，for linear operator theory，we prove the three famous theorems of bounded linear operators on Banach spaces： open mapping theorem，inverse operator theorem and closed graph theorems，moreover，we prove the uniform boundness principle (resonance theorem)，and analyze those excellent ideas，methods and proofs. Thirdly，spectrum theory of bounded linear operators plays a role in many scientific areas.It is an important content of the functional analysis，we list carefully basic concepts and useful properties，with enlightening examples. At the end of this chapter，on linear functional theory，we discuss mainly about the conjugate spaces of Banach and Hilbert spaces，as well as the conjugate operators of bounded linear operators in both spaces，including the famous Hermitian operator.
Distribution theory，a quite new direction in the crossdiscipline of scientific areas，appeared and was completed at the 1950s， spread out in Chapter 5.From Fourier transform of Lp(R)，1≤p≤2，Fourier transform of Schwartz function class，up to Fourier transform of Schwartz distributions，we arrange these contents in details. At the end of this chapter，the newest development of harmonic analysis — wavelet transform，is displayed with multisignal analysis and applied algorithms.The significance of this chapter is to recognize Fourier transform from the point of view of “distribution theory” in height.We emphasize that the famous Dirac δ function in physics turns out to be a distribution with compact support in seminormed distribution space ER，and this δ is the unit element of normed operator algebra LpR， ，α·，‖·‖LpR，，but δLpR，1≤p≤∞.It is certain that the new idea and new results in distribution theory could bring a new sense and effect to the “δfunction”， elegant and mystical，and has puzzled peoples mind for a long time.
Calculus on manifolds，not only the theme of our book，but also the essential base of differential geometry and Riemann geometry，organized in Chapter 6.Taking materials from ［3］ by the great master of mathematics，S. S. Chern，we start from basic concepts of smooth manifold，cotangent space，tangent space，vector field，tensor algebra，to exterior differential form on an exterior differential form space，in detail.Then，we show the definitions of exterior differentiation of an exterior differential form，and integration of an exterior differential form on a directed smooth manifold.Applied examples in Euclidean space as patterns are pointed out.The contents and concepts in this chapter are highly abstract and quite difficult to understand.We give models by threedimensional Euclidean space to help readers for establishing deepgoing and essential mathematics thought： from special cases to general ones，from concrete cases to abstract ones，from finite cases to infinite ones，and from theoretical cases to applied ones.These are the essence of modern applied mathematics.
Complimentary knowledge，disposed in the last chapter，including the useful variational calculus and some important theorems，such as StoneWeierstrass theorem，implicit mapping theorem，inverse mapping theorem，as well as the fixed point theorem on Banach spaces.Moreover，the Haar integral is introduced since it is needed in many natural science areas.
This is a selfcontained textbook with a wide span of knowledge，and its contents cover almost all of modern applied mathematics needed by research works in various natural sciences.
We have committed for several years to modify and replenish our teaching materials by practice，and to inspire abstract ideas by thinking in terms of images，to analyze difficult concepts by geometric ocular demonstration，to arrange certain questions and exercises for deepening derstanding.All these efforts are fruitful，thus， forming a complete textbook used independently for undergraduates，or as reference materials for other readers concerned.
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