Concise calculus:
Gespeichert in:
1. Verfasser: | |
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Weitere Verfasser: | |
Format: | Buch |
Sprache: | English |
Veröffentlicht: |
New Jersey
World Scientific
[2017]
|
Schlagworte: | |
Online-Zugang: | Inhaltsverzeichnis |
Beschreibung: | xvi, 674 Seiten Diagramme |
ISBN: | 9789814291484 9789814291491 |
Internformat
MARC
LEADER | 00000nam a22000002c 4500 | ||
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035 | |a (DE-599)BSZ372576532 | ||
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100 | 1 | |a Gong, Sheng |d 1930- |e Verfasser |0 (DE-588)1050696786 |4 aut | |
245 | 1 | 0 | |a Concise calculus |c Sheng Gong, University of Science & Technology of China, China ; translated by Youhong Gong |
264 | 1 | |a New Jersey |b World Scientific |c [2017] | |
300 | |a xvi, 674 Seiten |b Diagramme | ||
336 | |b txt |2 rdacontent | ||
337 | |b n |2 rdamedia | ||
338 | |b nc |2 rdacarrier | ||
650 | 0 | 7 | |a Analysis |0 (DE-588)4001865-9 |2 gnd |9 rswk-swf |
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700 | 1 | |a Gong, Youhong |4 trl | |
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999 | |a oai:aleph.bib-bvb.de:BVB01-025892271 |
Datensatz im Suchindex
_version_ | 1804150204821143552 |
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adam_text | Contents
Preface vii
1. Basic Concepts 1
1.1 Functions and Limits............................................ 1
1.1.1 Limits of Sequences and Functions....................... 1
1.1.2 Continuous Functions.................................... 2
1.2 Definite Integrals.............................................. 7
1.2.1 Calculation of Areas.................................... 7
1.2.2 Definition of Definite Integrals....................... 11
1.2.3 Logarithmic Function................................... 18
1.3 Derivatives and Differentials.................................. 24
1.3.1 Tangent Lines of a Curve............................... 24
1.3.2 Velocity and Density................................... 25
1.3.3 Definition of Derivatives ............................ 27
1.3.4 Differentials.......................................... 31
1.3.5 Mean-Value Theorem..................................... 33
1.4 Fundamental Theorem of Calculus................................ 38
2. Calculations of Derivatives and Integrals 43
2.1 Differentiation ............................................... 43
2.1.1 Calculations of Derivatives and Differentials ... 43
2.1.2 Derivatives and Differentials of Higher Order ... 54
2.1.3 Approximate Calculation by Derivatives ......... 58
2.2 Integration.................................................... 67
2.2.1 Indefinite Integrals .................................. 67
2.2.2 Definite Integrals..................................... 86
xi
93
101
101
101
103
106
111
112
114
115
118
120
126
126
131
141
149
149
149
152
161
167
167
171
181
190
196
209
209
209
211
Concise Calculus
2.2.3 Approximate Calculations of Definite Integrals . .
Some Applications of Differentiation and Integration
3.1 Areas, Volumes, Arc Lengths.................
3.1.1 Areas................................
3.1.2 Volumes .............................
3.1.3 Arc Lengths..........................
3.2 Techniques for Graphing Functions...........
3.2.1 Increasing and Decreasing Functions . .
3.2.2 Concavity ...........................
3.2.3 Asymptotes ..........................
3.2.4 Examples of Graphing Functions ....
3.2.5 Curvatures...........................
3.3 Taylor Expansions and Extreme Value Problems
3.3.1 Taylor Expansions....................
3.3.2 Extreme Value Problems...............
3.4 Examples in Physics.........................
Ordinary Differential Equations
4.1
4.2
First Order Differential Equations.........................
4.1.1 Concepts...........................................
4.1.2 Separation of Variables............................
4.1.3 Linear Differential Equations......................
Second Order Differential Equations........................
4.2.1 Reducible Differential Equations...................
4.2.2 Second Order Linear Differential Equations . . . .
4.2.3 Linear Differential Equations with Constant
Coefficients.......................................
4.2.4 Mechanical Vibration...............................
4.2.5 General Linear Differential Equations and Systems
of Linear Equations................................
Vector Algebra and Analytic Geometry in Three-
Dimensional Space
5.1 Coordinate System of Three-Dimensional Space and
Concept of Vectors...................................
5.1.1 Rectangular Coordinate System.................
5.1.2 Addition and Scalar Multiplication of Vectors . .
Contents
xiii
5.2 Products of Vectors ...................................... 217
5.2.1 Inner Products of Vectors......................... 217
5.2.2 Cross Products of Vectors ........................ 220
5.2.3 Scalar Triple Products of Vectors................. 222
5.3 Planes and Lines.......................................... 226
5.3.1 Equations of Planes............................... 226
5.3.2 Equations of Lines................................ 229
5.4 Quadric Surfaces.......................................... 234
5.4.1 Cylindrical Surfaces.............................. 234
5.4.2 Surfaces of Revolution ........................... 236
5.4.3 Conical Surfaces.................................. 238
5.4.4 Ellipsoid......................................... 239
5.4.5 Hyperbolic Paraboloid ............................ 241
5.4.6 Hyperboloid of One Sheet ......................... 242
5.4.7 Hyperboloid of Two Sheets......................... 242
5.4.8 Elliptic Paraboloid .............................. 242
5.5 Transformations of Coordinates............................ 244
5.5.1 Translation of Axes............................... 244
5.5.2 Rotation of Axes.................................. 246
6. Multiple Integrals and Partial Derivatives 251
6.1 Multiple Integrals........................................ 251
6.1.1 Limits and Continuity of Functions of Several
Variables......................................... 251
6.1.2 Multiple Integration.............................. 253
6.1.3 Calculation of Multiple Integrals................. 257
6.2 Partial Derivatives....................................... 270
6.2.1 Partial Derivatives and Total Differentials .... 270
6.2.2 Derivatives of Implicit Functions................. 278
6.3 Jacobian Determinants, Area Elements, Volume Elements 295
6.3.1 Properties of Jacobian Determinant................ 295
6.3.2 Area Elements and Volume Elements................. 296
7. Line Integrals, Surface Integrals and Exterior Differential Forms 317
7.1 Scalar Fields and Vector Fields........................... 317
7.1.1 Contour Surfaces and Gradient of a Scalar Field . 317
7.1.2 Streamlines of Vector Fields...................... 321
7.2 Line Integrals............................................ 327
xiv Concise Calculus
7.2.1 Line Integrals of the First Kind............... 327
7.2.2 Applications of Line Integrals of the First Kind
(Areas of Surfaces of Revolution) ............... 330
7.2.3 Line Integrals of the Second Kind.............. 331
7.2.4 Calculation of Line Integrals of the Second Kind . 335
7.2.5 Relation Between Line Integrals of the First Kind
and the Second Kind............................ 338
7.2.6 Circulations of Vector Fields and Line Integrals of
Vectors........................................ 339
7.3 Surface Integrals.......................................... 345
7.3.1 Surface Integrals of the First Kind............ 345
7.3.2 Flux of Vector Fields, Surface Integrals of the Sec-
ond Kind (Integral with respect to the projections
of the area element)........................... 348
7.3.3 Calculation of Surface Integrals of the Second Kind 350
7.4 Stokes Theorem............................................. 357
7.4.1 Green’s Theorem ................................... 357
7.4.2 Gauss’s Theorem, Divergence ....................... 361
7.4.3 Stokes’ Theorem, and The Curl of a Vector Field 367
7.5 Total Differentials and Line Integrals..................... 377
7.5.1 Line Integrals that are Independent of Paths . . . 377
7.5.2 Potential Fields............................... 381
7.5.3 Solenoidal Vector Fields....................... 383
7.6 Exterior Differential Forms................................ 387
7.6.1 Exterior Products, Exterior Differential Forms . . 387
7.6.2 Exterior Differentiation, Poincare Lemma and its
Inverse............................................ 394
7.6.3 Mathematical Meaning of Gradient, Divergence
and Curl........................................... 399
7.6.4 Fundamental Theorem of Calculus in Several
Variables (Stokes’ Theorem)........................ 401
8. Some Applications of Calculus in Several Variables 405
8.1 Taylor Expansions and Extremal Problems ................... 405
8.1.1 Taylor Expansions of Functions in Several
Variables...........................................405
8.1.2 Extremal Problems of Functions in Several
Variables......................................... 406
Conditional Extremum Problems .....................411
8.1.3
Contents XV
8.2 Examples of Applications in Physics........................417
8.2.1 B ary center, Moment of Inertia and Gravitational
Force............................................. 417
8.2.2 Complete System of Equations of Fluid Dynamics 423
8.2.3 Propagation of Sound...............................426
8.2.4 Heat Exchange......................................427
9. The e-S Definitions of Limits 433
9.1 The e-N Definition of Limits of Number Sequences .... 433
9.1.1 Definition of Limits of Number Sequences...........433
9.1.2 Properties of Limits of Number Sequences .... 435
9.1.3 Criteria for the Existence of Limits...............438
9.2 The e֊S Definition of Continuity of Functions..............447
9.2.1 Limits of Functions............................... 447
9.2.2 Definition of Continuous Functions ................454
9.2.3 Properties of Continuous Functions.................457
9.2.4 Uniform Continuity of Functions................... 460
9.3 Existence of Definite Integrals............................466
9.3.1 Darboux Sums...................................... 466
9.3.2 Integrability of Continuous Functions..............468
9.3.3 Generalization of the Concept of Definite Integrals
(Improper Integrals)...............................475
10. Infinite Series and Infinite Integrals 485
10.1 Number Series ............................................ 485
10.1.1 Basic Concepts.................................... 485
10.1.2 Some Convergence Criteria......................... 487
10.1.3 Conditionally Convergent Series................... 493
10.2 Function Series........................................... 502
10.2.1 Infinite Sums..................................... 502
10.2.2 Uniformly Convergent Sequences of Functions . . 504
10.2.3 Uniformly Convergent Function Series.............. 508
10.2.4 Existence Theorem of Implicit Functions........... 512
10.2.5 Existence and Uniqueness Theorem of the Solution
of Ordinary Differential Equations................ 516
10.3 Power Series and Taylor Series............................ 524
10.3.1 Convergence Radius of Power Series................ 524
10.3.2 Properties of Power Series........................ 527
XVI
Concise Calculus
10.3.3 Taylor Series................................... 532
10.3.4 Applications of Power Series.................... 539
10.4 Infinite Integrals and Integrals with Parameters........ 553
10.4.1 Convergence Criteria for Infinite Integrals......553
10.4.2 Integrals with Parameters....................... 565
10.4.3 Infinite Integrals with Parameters.............. 570
10.4.4 Several Important Infinite Integrals............ 583
11. Fourier Series and Fourier Integrals 597
11.1 Fourier Series.......................................... 597
11.1.1 Orthogonality of the System of Trigonometric
Functions....................................... 597
11.1.2 Bessel Inequality............................... 607
11.1.3 Convergence Criterion for Fourier Series ........611
11.2 Fourier Integrals....................................... 616
11.2.1 Fourier Integrals............................... 616
11.2.2 Fourier Transforms.............................. 619
11.2.3 Applications of Fourier Transforms.............. 624
11.2.4 Higher-Dimensional Fourier Transforms........... 625
Answers 627
Index 671
|
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author | Gong, Sheng 1930- |
author2 | Gong, Youhong |
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author_GND | (DE-588)1050696786 |
author_facet | Gong, Sheng 1930- Gong, Youhong |
author_role | aut |
author_sort | Gong, Sheng 1930- |
author_variant | s g sg |
building | Verbundindex |
bvnumber | BV040912979 |
classification_rvk | SK 400 |
classification_tum | MAT 260f |
ctrlnum | (OCoLC)991542494 (DE-599)BSZ372576532 |
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id | DE-604.BV040912979 |
illustrated | Not Illustrated |
indexdate | 2024-07-10T00:35:13Z |
institution | BVB |
isbn | 9789814291484 9789814291491 |
language | English |
oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025892271 |
oclc_num | 991542494 |
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owner | DE-29T DE-20 DE-703 DE-739 DE-91G DE-BY-TUM |
owner_facet | DE-29T DE-20 DE-703 DE-739 DE-91G DE-BY-TUM |
physical | xvi, 674 Seiten Diagramme |
publishDate | 2017 |
publishDateSearch | 2017 |
publishDateSort | 2017 |
publisher | World Scientific |
record_format | marc |
spelling | Gong, Sheng 1930- Verfasser (DE-588)1050696786 aut Concise calculus Sheng Gong, University of Science & Technology of China, China ; translated by Youhong Gong New Jersey World Scientific [2017] xvi, 674 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Analysis (DE-588)4001865-9 gnd rswk-swf Analysis (DE-588)4001865-9 s DE-604 Gong, Youhong trl Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025892271&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
spellingShingle | Gong, Sheng 1930- Concise calculus Analysis (DE-588)4001865-9 gnd |
subject_GND | (DE-588)4001865-9 |
title | Concise calculus |
title_auth | Concise calculus |
title_exact_search | Concise calculus |
title_full | Concise calculus Sheng Gong, University of Science & Technology of China, China ; translated by Youhong Gong |
title_fullStr | Concise calculus Sheng Gong, University of Science & Technology of China, China ; translated by Youhong Gong |
title_full_unstemmed | Concise calculus Sheng Gong, University of Science & Technology of China, China ; translated by Youhong Gong |
title_short | Concise calculus |
title_sort | concise calculus |
topic | Analysis (DE-588)4001865-9 gnd |
topic_facet | Analysis |
url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025892271&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
work_keys_str_mv | AT gongsheng concisecalculus AT gongyouhong concisecalculus |