Advanced calculus: theory and practice
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
CRC Press
2014
|
| Series: | Textbooks in mathematics
A Chapman & Hall book |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | XII, 560 S. |
| ISBN: | 9781466565630 |
Staff View
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Record in the Search Index
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| adam_text | Contents
B
Preface ix
1 Sequences and Their Limits 1
1.1 Computing the Limits: Part I ........................................... 1
1.2 Definition of the Limit ................................................ 4
1.3 Properties of Limits .......................................*....... 8
1.4 Monotone Sequences..................................................... 13
1.5 Number e............................................................... 17
1.6 Cauchy Sequences ...................................................... 22
1.7 Limit Superior and Limit Inferior ..................................... 25
1.8 Computing the Limits: Part II ......................................... 29
2 Real Numbers 35
2.1 Axioms of the Set M.................................................... 35
2.2 Consequences of the Completeness Axiom................................. 39
2.3 Bolzano-Weierstrass Theorem ........................................... 44
2.4 Some Thoughts about M ................................................. 47
3 Continuity 51
3.1 Computing Limits of Functions......................................... 51
3.2 Review of Functions.................................................... 56
3.3 Continuous Functions: A Geometric Viewpoint............................ 58
3.4 Limits of Functions ................................................... 61
3.5 Other Limits........................................................... 66
3.5.1 One-Sided Limits................................................ 66
3.5.2 Limits at Infinity.............................................. 68
3.5.3 Infinite Limits ................................................ 70
3.6 Properties of Continuous Functions .................................... 72
3.7 Continuity of Elementary Functions .................................... 77
3.8 Uniform Continuity .................................................. 81
3.9 Two Properties of Continuous Functions ................................ 85
4 Derivative 91
4.1 Computing the Derivatives .................................................... 91
4.2 Derivative ................................................................... 93
4.3 Rules of Differentiation ..................................................... 99
4.4 Monotonicity: Local Extrema.................................................. 103
4.5 Taylor’s Formula............................................................. 109
v
vi
4.6 L’HôpitaPs Rule ........................................................ 113
5 Indefinite Integral 117
5.1 Computing Indefinite Integrals.......................................... 117
5.2 Antiderivative ......................................................... 121
5.2.1 Rational Functions............................................... 124
5.2.2 Irrational Functions............................................. 127
5.2.3 Binomial Differentials........................................... 128
5.2.4 Some Trigonometric Integrals..................................... 131
6 Definite Integral 135
6.1 Computing Definite Integrals............................................ 135
6.2 Definite Integral ...................................................... 138
6.3 Integrable Functions.................................................... 143
6.4 Riemann Sums............................................................ 148
6.5 Properties of Definite Integrals........................................ 152
6.6 Fundamental Theorem of Calculus......................................... 155
6.7 Infinite and Improper Integrals......................................... 159
6.7.1 Infinite Integrals............................................... 159
6.7.2 Improper Integrals............................................... 163
7 Infinite Series 169
7.1 Review of Infinite Series.............................................. 169
7.2 Definition of a Series ................................................. 173
7.3 Series with Positive Terms ............................................. 176
7.4 Root and Ratio Tests ................................................... 181
7.4.1 Additional Tests for Convergence ................................ 183
7.5 Series with Arbitrary Terms ............................................ 186
7.5.1 Additional Tests for Convergence................................. 189
7.5.2 Rearrangement of a Series........................................ 192
8 Sequences and Series of Functions 197
8.1 Convergence of a Sequence of Functions.................................. 197
8.2 Uniformly Convergent Sequences of Functions ........................... 202
8.3 Function Series......................................................... 208
8.3.1 Applications to Differential Equations........................... 211
8.3.2 Continuous Nowhere Differentiable Function ...................... 214
8.4 Power Series ........................................................... 216
8.5 Power Series Expansions of Elementary Functions ...................... 221
9 Fourier Series 229
9.1 Introduction ........................................................... 229
9.2 Pointwise Convergence of Fourier Series................................. 233
9.3 Uniform Convergence of Fourier Series .................................. 239
9.4 Cesàro Summability...................................................... 244
9.5 Mean Square Convergence of Fourier Series.............................. . 250
Vll
9.6 Influence of Fourier Series............................................. 255
10 Functions of Several Variables 259
10.1 Subsets of W1 .......................................................... 259
10.2 Functions and Their Limits............................................ 264
10.3 Continuous Functions.................................................... 269
10.4 Boundedness of Continuous Functions..................................... 273
10.5 Open Sets in Rn......................................................... 278
10.6 Intermediate Value Theorem.............................................. 285
10.7 Compact Sets ........................................................... 291
11 Derivatives 297
11.1 Computing Derivatives .................................................. 297
11.2 Derivatives and Differentiability ............................-........ 300
11.3 Properties of the Derivative............................................ 306
11.4 Functions from Rn to Mm................................................. 310
11.5 Taylor’s Formula....................................................... 314
11.6 Extreme Values ......................................................... 318
12 Implicit Functions and Optimization 325
12.1 Implicit Functions ..................................................... 325
12.2 Derivative as a Linear Map.............................................. 330
12.3 Open Mapping Theorem ................................................... 335
12.4 Implicit Function Theorem............................................... 339
12.5 Constrained Optimization................................................ 344
12.6 Second Derivative Test ................................................. 351
12.6.1 Absolute Extrema.................................................. 355
13 Integrals Depending on a Parameter 357
13.1 Uniform Convergence .................................................... 357
13.2 Integral as a Function................................................. 361
13.3 Uniform Convergence of Improper Integrals.............................. 367
13.4 Integral as a Function.................................................. 371
13.5 Some Important Integrals................................................ 377
14 Integration in Rn 387
14.1 Double Integrals over Rectangles........................................ 387
14.2 Double Integrals over Jordan Sets ..................................... 393
14.3 Double Integrals as Iterated Integrals ................................. 397
14.4 Transformations of Jordan Sets in R2 .................................. 402
14.5 Change of Variables in Double Integrals................................ 407
14.6 Improper Integrals...................................................... 413
14.7 Multiple Integrals ..................................................... 418
Vlll
15 Fundamental Theorems 425
15.1 Curves in W1........................................................... 425
15.2 Line Integrals ........................................................ 430
15.3 Green’s Theorem ....................................................... 434
15.4 Surface Integrals...................................................... 439
15.5 Divergence Theorem..................................................... 446
15.6 Stokes’ Theorem...................................................... 449
15.7 Differential Forms on Rn .......................................... 453
15.8 Exact Differential Forms on Rn ........................................ 457
16 Solutions and Answers to Selected Problems 465
Bibliography 543
Subject Index 551
Author Index
559
Mathematics
TEXTBOOKS in MATHEMATICS
ADVANCED
CALCULUS
THEORY AND PRACTICE
Advanced Calculus: Theory and Practice expands on the material covered in
elementary calculus and presents this material in a rigorous manner. The text
improves your problem-solving and proof-writing skills, familiarizes you with
the historical development of calculus concepts, and helps you understand the
connections among different topics.
The book takes a motivating approach that makes ideas less abstract. It explains
how various topics in calculus may seem unrelated but in reality have common
roots. Emphasizing historical perspectives, the text gives you a glimpse into the
development of calculus and its ideas from the age of Newton and Leibniz to the
twentieth century. Nearly 300 examples lead to important theorems as well as
help you develop the necessary skills to closely examine the theorems. Proofs
are also presented in an accessible way.
Features
• Presents clear proofs that illuminate the context of the theorems
• Points out the relationship among various topics in calculus
• Includes a substantial amount of history, a large number of motivating
examples, and an extensive list of references for further research of results
or the history of a concept
• Contains almost 100 worked-out exercises and over 1,000 homework
problems to develop your problem-solving skills
By strengthening skills gained through elementary calculus, this textbook leads
you toward mastering calculus techniques. It will help you succeed in your future
mathematical or engineering studies.
ISBN: ‘Dfl-l-MbbS-bSbi-O
|
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| spelling | Petrovic, John Srdjan Verfasser aut Advanced calculus theory and practice John Srdjan Petrovic Boca Raton [u.a.] CRC Press 2014 XII, 560 S. txt rdacontent n rdamedia nc rdacarrier Textbooks in mathematics A Chapman & Hall book MATHEMATICS / Applied bisacsh MATHEMATICS / Differential Equations bisacsh MATHEMATICS / Functional Analysis bisacsh Calculus Textbooks MATHEMATICS / Applied MATHEMATICS / Differential Equations MATHEMATICS / Functional Analysis Analysis (DE-588)4001865-9 gnd rswk-swf Analysis (DE-588)4001865-9 s DE-604 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027709476&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027709476&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Petrovic, John Srdjan Advanced calculus theory and practice MATHEMATICS / Applied bisacsh MATHEMATICS / Differential Equations bisacsh MATHEMATICS / Functional Analysis bisacsh Calculus Textbooks MATHEMATICS / Applied MATHEMATICS / Differential Equations MATHEMATICS / Functional Analysis Analysis (DE-588)4001865-9 gnd |
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| title | Advanced calculus theory and practice |
| title_auth | Advanced calculus theory and practice |
| title_exact_search | Advanced calculus theory and practice |
| title_full | Advanced calculus theory and practice John Srdjan Petrovic |
| title_fullStr | Advanced calculus theory and practice John Srdjan Petrovic |
| title_full_unstemmed | Advanced calculus theory and practice John Srdjan Petrovic |
| title_short | Advanced calculus |
| title_sort | advanced calculus theory and practice |
| title_sub | theory and practice |
| topic | MATHEMATICS / Applied bisacsh MATHEMATICS / Differential Equations bisacsh MATHEMATICS / Functional Analysis bisacsh Calculus Textbooks MATHEMATICS / Applied MATHEMATICS / Differential Equations MATHEMATICS / Functional Analysis Analysis (DE-588)4001865-9 gnd |
| topic_facet | MATHEMATICS / Applied MATHEMATICS / Differential Equations MATHEMATICS / Functional Analysis Calculus Textbooks Analysis |
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