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E-raamat: Multivariable Calculus and Differential Geometry [De Gruyter e-raamatud]

  • Formaat: 365 pages
  • Sari: De Gruyter Textbook
  • Ilmumisaeg: 12-Jun-2015
  • Kirjastus: De Gruyter
  • ISBN-13: 9783110369540
Teised raamatud teemal:
  • De Gruyter e-raamatud
  • Hind: 540,00 €*
  • * hind, mis tagab piiramatu üheaegsete kasutajate arvuga ligipääsu piiramatuks ajaks
Multivariable Calculus and Differential GeometrySuurem pilt
  • Formaat: 365 pages
  • Sari: De Gruyter Textbook
  • Ilmumisaeg: 12-Jun-2015
  • Kirjastus: De Gruyter
  • ISBN-13: 9783110369540
Teised raamatud teemal:
Wiley OnlineRohkem infot De Gruyter e-raamatute kohta
This book offers an introduction to differential geometry for the non-specialist. It includes most of the required material from multivariable calculus, linear algebra, and basic analysis. An intuitive approach and a minimum of prerequisites make it a valuable companion for students of mathematics and physics.

The main focus is on manifolds in Euclidean space and the metric properties they inherit from it. Among the topics discussed are curvature and how it affects the shape of space, and the generalization of the fundamental theorem of calculus known as Stokes' theorem.
Preface v
1 Euclidean Space
1(56)
1.1 Vector spaces
1(5)
1.2 Linear transformations
6(6)
1.3 Determinants
12(7)
1.4 Euclidean spaces
19(6)
1.5 Subspaces of Euclidean space
25(2)
1.6 Determinants as volume
27(3)
1.7 Elementary topology of Euclidean spaces
30(6)
1.8 Sequences
36(5)
1.9 Limits and continuity
41(7)
1.10 Exercises
48(9)
2 Differentiation
57(60)
2.1 The derivative
57(5)
2.2 Basic properties of the derivative
62(5)
2.3 Differentiation of integrals
67(2)
2.4 Curves
69(6)
2.5 The inverse and implicit function theorems
75(6)
2.6 The spectral theorem and scalar products
81(8)
2.7 Taylor polynomials and extreme values
89(5)
2.8 Vector fields
94(9)
2.9 Lie brackets
103(5)
2.10 Partitions of unity
108(2)
2.11 Exercises
110(7)
3 Manifolds
117(60)
3.1 Submanifolds of Euclidean space
117(7)
3.2 Differentiable maps on manifolds
124(5)
3.3 Vector fields on manifolds
129(8)
3.4 Lie groups
137(4)
3.5 The tangent bundle
141(2)
3.6 Covariant differentiation
143(5)
3.7 Geodesies
148(5)
3.8 The second fundamental tensor
153(3)
3.9 Curvature
156(4)
3.10 Sectional curvature
160(3)
3.11 Isometries
163(5)
3.12 Exercises
168(9)
4 Integration on Euclidean space
177(44)
4.1 The integral of a function over a box
177(4)
4.2 Integrability and discontinuities
181(6)
4.3 Fubini's theorem
187(8)
4.4 Sard's theorem
195(3)
4.5 The change of variables theorem
198(4)
4.6 Cylindrical and spherical coordinates
202(8)
4.6.1 Cylindrical coordinates
202(4)
4.6.2 Spherical coordinates
206(4)
4.7 Some applications
210(4)
4.7.1 Mass
211(1)
4.7.2 Center of mass
211(2)
4.7.3 Moment of inertia
213(1)
4.8 Exercises
214(7)
5 Differential Forms
221(46)
5.1 Tensors and tensor fields
221(3)
5.2 Alternating tensors and forms
224(8)
5.3 Differential forms
232(4)
5.4 Integration on manifolds
236(4)
5.5 Manifolds with boundary
240(3)
5.6 Stokes' theorem
243(3)
5.7 Classical versions of Stokes' theorem
246(6)
5.7.1 An application: the polar planimeter
249(3)
5.8 Closed forms and exact forms
252(5)
5.9 Exercises
257(10)
6 Manifolds as metric spaces
267(34)
6.1 Extremal properties of geodesies
267(4)
6.2 Jacobi fields
271(4)
6.3 The length function of a variation
275(3)
6.4 The index form of a geodesic
278(5)
6.5 The distance function
283(2)
6.6 The Hopf-Rinow theorem
285(4)
6.7 Curvature comparison
289(3)
6.8 Exercises
292(9)
7 Hypersurfaces
301(38)
7.1 Hypersurfaces and orientation
301(3)
7.2 The Gauss map
304(4)
7.3 Curvature of hypersurfaces
308(5)
7.4 The fundamental theorem for hypersurfaces
313(3)
7.5 Curvature in local coordinates
316(2)
7.6 Convexity and curvature
318(2)
7.7 Ruled surfaces
320(3)
7.8 Surfaces of revolution
323(5)
7.9 Exercises
328(11)
Appendix A 339(6)
Appendix B 345(6)
Index 351
Gerard Walschap, University of Oklahoma, Norman, OK, USA.
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