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E-raamat: Several Real Variables

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Several Real VariablesSuurem pilt
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This undergraduate textbook is based on lectures given by the author on the differential and integral calculus of functions of several real variables. The book has a modern approach and includes topics such as:

The p-norms on vector space and their equivalence

The Weierstrass and Stone-Weierstrass approximation theorems

The differential as a linear functional; Jacobians, Hessians, and Taylor's theorem in several variables

The Implicit Function Theorem for a system of equations, proved via Banachs Fixed Point Theorem

Applications to Ordinary Differential Equations

Line integrals and an introduction to surface integrals

This book features numerous examples, detailed proofs, as well as exercises at the end of sections. Many of the exercises have detailed solutions, making the book suitable for self-study.





Several Real Variables will be useful for undergraduate students in mathematics who have completed first courses in linear algebra and analysis of one real variable.

Arvustused

This textbook is excellent for self-study as it contains a lot of well-chosen examples and complete proofs. Each topic is followed by a set of suitable exercises, accompanied with detailed solutions, which complement the presented material significantly. The reader can obtain a very deep insight into analysis and related theories as the book provides deep understanding of principles. (Roza Aceska, zbMATH 1405.26003, 2019)

1 Continuity
1(58)
1.1 The Normed Space Rk
1(23)
Inner Product Space
2(1)
Normed Space
2(2)
Holder's Inequality
4(1)
Minkowski's Inequality
5(1)
The Norm ||·||∞ on Rk
6(1)
Equivalence of the Norms ||·||p
6(1)
Examples
7(3)
Metric
10(1)
The Metric Topology
11(5)
Characterization of Closed Sets
16(2)
Connected Sets
18(2)
Union of Connected Sets
20(2)
Sufficient Condition for Connectedness
22(1)
Exercises
22(2)
1.2 Compact Sets
24(7)
Open Cover and Compactness
25(1)
Properties of Compact Sets
25(3)
Compactness of Closed Bounded Sets in Rk
28(2)
Exercises
30(1)
1.3 Sequences
31(9)
Basics
31(2)
Subsequences
33(1)
Cauchy Sequences
34(3)
Exercises
37(3)
1.4 Functions
40(19)
Basics
40(2)
Limits
42(2)
Continuous Functions
44(3)
Properties of Continuous Functions
47(2)
Uniform Continuity
49(3)
Paths
52(2)
Domains in Rk
54(1)
Exercises
55(4)
2 Derivation
59(36)
2.1 Differentiability
59(20)
Directional Derivatives
59(4)
The Differential
63(6)
Examples
69(3)
The Chain Rule
72(3)
The Differential of a Vector Valued Function
75(2)
Exercises
77(2)
2.2 Higher Derivatives
79(16)
Mixed Derivatives
79(5)
Taylor's Theorem
84(2)
Local Extrema
86(3)
The Second Differential
89(2)
Exercises
91(4)
3 Implicit Functions
95(68)
3.1 Fixed Points
95(5)
The Banach Fixed Point Theorem
96(1)
The Space C(X)
97(3)
3.2 The Implicit Function Theorem
100(7)
Lipschitz' Condition
100(3)
The Implicit Function Theorem
103(3)
Exercises
106(1)
3.3 System of Equations
107(8)
The Implicit Function Theorem for Systems
108(5)
The Local Inverse Map Theorem
113(1)
The Jacobian of a Composed Map
113(2)
Exercises
115(1)
3.4 Extrema with Constraints
115(5)
Exercises
119(1)
3.5 Applications in R3
120(12)
Surfaces
120(8)
Tangent Plane
128(2)
Exercises
130(2)
3.6 Application to Ordinary Differential Equations
132(18)
Existence and Uniqueness Theorem
133(4)
Linear ODE
137(2)
Fundamental Matrix
139(2)
Exercises
141(9)
3.7 More on C(I)
150(13)
The Weierstrass Approximation Theorem
152(2)
The Arzela-Ascoli Theorem
154(4)
The Stone-Weierstrass Theorem
158(3)
Exercises
161(2)
4 Integration
163(78)
4.1 Partial Integrals
163(14)
Variable Limits of Integration
168(2)
General Leibnitz' Rule
170(2)
Changing the Order of Integration
172(3)
Exercises
175(2)
4.2 Integration on a Domain in Rk
177(30)
Content
178(2)
Integration on a Bounded Closed Domain in Rk
180(3)
Basic Properties of R(D)
183(4)
Multiple Integrals as Iterated Integrals
187(4)
Normal Domains
191(2)
Change of Variables
193(1)
Examples
194(10)
Integration on Unbounded Domains in Rk
204(2)
Exercises
206(1)
4.3 Line Integrals
207(16)
Smooth Curves
210(3)
Line Integrals
213(2)
Conservative Fields
215(2)
Necessary Condition
217(2)
Sufficient Condition
219(2)
Locally Conservative Fields
221(1)
Exercises
221(2)
4.4 Green's Theorem in R2
223(6)
Green's Theorem for Normal Domains
224(3)
General Green's Theorem in R2
227(1)
Exercises
228(1)
4.5 Surface Integrals in R3
229(12)
Surface Area
229(1)
Surface Integral
230(1)
Flux of a Vector Field Through a Surface
231(2)
The Divergence Theorem
233(2)
Stokes' Formula
235(2)
Exercises
237(4)
Appendix A Solutions 241(62)
References 303(2)
Index 305
The author is an expert in Spectral Theory and semigroups of operators, researching in this area for 50 years, publishing numerous papers and three monographs.  
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