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Introduction to Analysis in Several Variables: Advanced Calculus [Minkštas viršelis]

  • Formatas: Paperback / softback, 440 pages, aukštis x plotis: 254x178 mm, weight: 812 g
  • Serija: Pure and Applied Undergraduate Texts
  • Išleidimo metai: 30-Sep-2020
  • Leidėjas: American Mathematical Society
  • ISBN-10: 1470456699
  • ISBN-13: 9781470456696
Kitos knygos pagal šią temą:
Introduction to Analysis in Several Variables: Advanced CalculusDidesnis paveikslėlis
  • Formatas: Paperback / softback, 440 pages, aukštis x plotis: 254x178 mm, weight: 812 g
  • Serija: Pure and Applied Undergraduate Texts
  • Išleidimo metai: 30-Sep-2020
  • Leidėjas: American Mathematical Society
  • ISBN-10: 1470456699
  • ISBN-13: 9781470456696
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781470456696.html
Raktažodžiai:
This text was produced for the second part of a two-part sequence on advanced calculus, whose aim is to provide a firm logical foundation for analysis. The first part treats analysis in one variable, and the text at hand treats analysis in several variables.

After a review of topics from one-variable analysis and linear algebra, the text treats in succession multivariable differential calculus, including systems of differential equations, and multivariable integral calculus. It builds on this to develop calculus on surfaces in Euclidean space and also on manifolds. It introduces differential forms and establishes a general Stokes formula. It describes various applications of Stokes formula, from harmonic functions to degree theory.

The text then studies the differential geometry of surfaces, including geodesics and curvature, and makes contact with degree theory, via the Gauss-Bonnet theorem. The text also takes up Fourier analysis, and bridges this with results on surfaces, via Fourier analysis on spheres and on compact matrix groups.
Preface vii
Some basic notation xi
Chapter 1 Background
1(38)
1.1 One-variable calculus
2(15)
1.2 Euclidean spaces
17(5)
1.3 Vector spaces and linear transformations
22(9)
1.4 Determinants
31(8)
Chapter 2 Multivariable differential calculus
39(48)
2.1 The derivative
39(17)
2.2 Inverse function and implicit function theorems
56(12)
2.3 Systems of differential equations and vector fields
68(19)
Chapter 3 Multivariable integral calculus and calculus on surfaces
87(66)
3.1 The Riemann integral in n variables
88(29)
3.2 Surfaces and surface integrals
117(28)
3.3 Partitions of unity
145(1)
3.4 Sard's theorem
146(1)
3.5 Morse functions
147(1)
3.6 The tangent space to a manifold
148(5)
Chapter 4 Differential forms and the Gauss-Green-Stokes formula
153(32)
4.1 Differential forms
154(6)
4.2 Products and exterior derivatives of forms
160(4)
4.3 The general Stokes formula
164(5)
4.4 The classical Gauss, Green, and Stokes formulas
169(10)
4.5 Differential forms and the change of variable formula
179(6)
Chapter 5 Applications of the Gauss-Green-Stokes formula
185(36)
5.1 Holomorphic functions and harmonic functions
186(14)
5.2 Differential forms, homotopy, and the Lie derivative
200(5)
5.3 Differential forms and degree theory
205(16)
Chapter 6 Differential geometry of surfaces
221(70)
6.1 Geometry of surfaces I: geodesies
225(13)
6.2 Geometry of surfaces II: curvature
238(14)
6.3 Geometry of surfaces III: the Gauss-Bonnet theorem
252(13)
6.4 Smooth matrix groups
265(18)
6.5 The derivative of the exponential map
283(5)
6.6 A spectral mapping theorem
288(3)
Chapter 7 Fourier analysis
291(90)
7.1 Fourier series
294(16)
7.2 The Fourier transform
310(20)
7.3 Poisson summation formulas
330(2)
7.4 Spherical harmonics
332(40)
7.5 Fourier series on compact matrix groups
372(6)
7.6 Isoperimetric inequality
378(4)
Appendix A Complementary material 381(56)
A.1 Metric spaces, convergence, and compactness
382(11)
A.2 Inner product spaces
393(5)
A.3 Eigenvalues and eigenvectors
398(4)
A.4 Complements on power series
402(6)
A.5 The Weierstrass theorem and the Stone-Weierstrass theorem
408(2)
A.6 Further results on harmonic functions
410(6)
A.7 Beyond degree theory---introduction to de Rham theory
416(21)
Bibliography 437(4)
Index 441
Michael E. Taylor, University of North Carolina, Chapel Hill, NC