A Course in Multivariable Calculus and Analysis

by Ghorpade, Sudhir R.; Limaye, Balmohan V.

• ISBN13: 9781441916204
• ISBN10: 1441916202
• Edition: 1st
• Format: Hardcover
• Copyright: 2009-11-23
• Publisher: Springer Nature

Summary

This self-contained textbook gives a thorough exposition of multivariable calculus. It can be viewed as a sequel to the one-variable calculus text, A Course in Calculus and Real Analysis, published in the same series. The emphasis is on correlating general concepts and results of multivariable calculus with their counterparts in one-variable calculus. For example, when the general definition of the volume of a solid is given using triple integrals, the authors explain why the shell and washer methods of one-variable calculus for computing the volume of a solid of revolution must give the same answer. Further, the book includes genuine analogues of basic results in one-variable calculus, such as the mean value theorem and the fundamental theorem of calculus.This book is distinguished from others on the subject: it examines topics not typically covered, such as monotonicity, bimonotonicity, and convexity, together with their relation to partial differentiation, cubature rules for approximate evaluation of double integrals, and conditional as well as unconditional convergence of double series and improper double integrals. Moreover, the emphasis is on a geometric approach to such basic notions as local extremum and saddle point. Each chapter contains detailed proofs of relevant results, along with numerous examples and a wide collection of exercises of varying degrees of difficulty, making the book useful to undergraduate and graduate students alike. There is also an informative section of "Notes and Comments'' indicating some novel features of the treatment of topics in that chapter as well as references to relevant literature. The only prerequisite for this text is a course in one-variable calculus.

Table of Contents

Vectors and Functions	p. 1
Preliminaries	p. 2
Algebraic Operations	p. 2
Order Properties	p. 4
Intervals, Disks, and Bounded Sets	p. 6
Line Segments and Paths	p. 8
Functions and Their Geometric Properties	p. 10
Basic Notions	p. 10
Bounded Functions	p. 13
Monotonicity and Bimonotonicity	p. 14
Functions of Bounded Variation	p. 17
Functions of Bounded Bivariation	p. 20
Convexity and Concavity	p. 25
Local Extrema and Saddle Points	p. 26
Intermediate Value Property	p. 29
Cylindrical and Spherical Coordinates	p. 30
Cylindrical Coordinates	p. 31
Spherical Coordinates	p. 32
Notes and Comments	p. 33
Exercises	p. 34
Sequences, Continuity, and Limits	p. 43
Sequences in R2	p. 43
Subsequences and Cauchy Sequences	p. 45
Closure, Boundary, and Interior	p. 46
Continuity	p. 48
Composition of Continuous Functions	p. 51
Piecing Continuous Functions on Overlapping Subsets	p. 53
Characterizations of Continuity	p. 55
Continuity and Boundedness	p. 56
Continuity and Monotonicity	p. 57
Continuity, Bounded Variation, and Bounded Bivariation	p. 57
Continuity and Convexity	p. 58
Continuity and Intermediate Value Property	p. 60
Uniform Continuity	p. 61
Implicit Function Theorem	p. 63
Limits	p. 67
Limits and Continuity	p. 68
Limit from a Quadrant	p. 71
Approaching Infinity	p. 72
Notes and Comments	p. 76
Exercises	p. 77
Partial and Total Differentiation	p. 83
Partial and Directional Derivatives	p. 84
Partial Derivatives	p. 84
Directional Derivatives	p. 88
Higher-Order Partial Derivatives	p. 91
Higher-Order Directional Derivatives	p. 99
Differentiability	p. 101
Differentiability and Directional Derivatives	p. 109
Implicit Differentiation	p. 112
Taylor's Theorem and Chain Rule	p. 116
Bivariate Taylor Theorem	p. 116
Chain Rule	p. 120
Monotonicity and Convexity	p. 125
Monotonicity and First Partials	p. 125
Bimonotonicity and Mixed Partials	p. 126
Bounded Variation and Boundedness of First Partials	p. 127
Bounded Bivariation and Boundedness of Mixed Partials	p. 128
Convexity and Monotonicity of Gradient	p. 129
Convexity and Nonnegativity of Hessian	p. 133
Functions of Three Variables	p. 138
Extensions and Analogues	p. 138
Tangent Planes and Normal Lines to Surfaces	p. 143
Convexity and Ternary Quadratic Forms	p. 147
Notes and Comments	p. 149
Exercises	p. 151
Applications of Partial Differentiation	p. 157
Absolute Extrema	p. 157
Boundary Points and Critical Points	p. 158
Constrained Extrema	p. 161
Lagrange Multiplier Method	p. 162
Case of Three Variables	p. 164
Local Extrema and Saddle Points	p. 167
Discriminant Test	p. 170
Linear and Quadratic Approximations	p. 175
Linear Approximation	p. 175
Quadratic Approximation	p. 178
Notes and Comments	p. 180
Exercises	p. 181
Multiple Integration	p. 185
Double Integrals on Rectangles	p. 185
Basic Inequality and Criterion for Integrability	p. 193
Domain Additivity on Rectangles	p. 197
Integrability of Monotonic and Continuous Functions	p. 200
Algebraic and Order Properties	p. 202
A Version of the Fundamental Theorem of Calculus	p. 208
Fubini's Theorem on Rectangles	p. 216
Riemann Double Sums	p. 222
Double Integrals over Bounded Sets	p. 226
Fubini's Theorem over Elementary Regions	p. 230
Sets of Content Zero	p. 232
Concept of Area of a Bounded Subset of R2	p. 240
Domain Additivity over Bounded Sets	p. 244
Change of Variables	p. 247
Translation Invariance and Area of a Parallelogram	p. 247
Case of Affine Transformations	p. 251
General Case	p. 258
Triple Integrals	p. 267
Triple Integrals over Bounded Sets	p. 269
Sets of Three-Dimensional Content Zero	p. 273
Concept of Volume of a Bounded Subset of R3	p. 273
Change of Variables in Triple Integrals	p. 274
Notes and Comments	p. 280
Exercises	p. 282
Applications and Approximations of Multiple Integrals	p. 291
Area and Volume	p. 291
Area of a Bounded Subset of R2	p. 291
Regions between Polar Curves	p. 293
Volume of a Bounded Subset of R3	p. 297
Solids between Cylindrical or Spherical Surfaces	p. 298
Slicing by Planes and the Washer Method	p. 302
Slivering by Cylinders and the Shell Method	p. 303
Surface Area	p. 309
Parallelograms in R2 and in R3	p. 311
Area of a Smooth Surface	p. 313
Surfaces of Revolution	p. 319
Centroids of Surfaces and Solids	p. 322
Averages and Weighted Averages	p. 323
Centroids of Planar Regions	p. 324
Centroids of Surfaces	p. 326
Centroids of Solids	p. 329
Centroids of Solids of Revolution	p. 335
Cubature Rules	p. 338
Product Rules on Rectangles	p. 339
Product Rules over Elementary Regions	p. 344
Triangular Prism Rules	p. 346
Notes and Comments	p. 360
Exercises	p. 361
Double Series and Improper Double Integrals	p. 369
Double Sequences	p. 369
Monotonicity and Bimonotonicity	p. 373
Convergence of Double Series	p. 376
Telescoping Double Series	p. 382
Double Series with Nonnegative Terms	p. 383
Absolute Convergence and Conditional Convergence	p. 387
Unconditional Convergence	p. 390
Convergence Tests for Double Series	p. 392
Tests for Absolute Convergence	p. 392
Tests for Conditional Convergence	p. 399
Double Power Series	p. 403
Taylor Double Series and Taylor Series	p. 411
Convergence of Improper Double Integrals	p. 416
Improper Double Integrals of Mixed Partials	p. 420
Improper Double Integrals of Nonnegative Functions	p. 421
Absolute Convergence and Conditional Convergence	p. 425
Convergence Tests for Improper Double Integrals	p. 428
Tests for Absolute Convergence	p. 430
Tests for Conditional Convergence	p. 431
Unconditional Convergence of Improper Double Integrals	p. 435
Functions on Unbounded Subsets	p. 436
Concept of Area of an Unbounded Subset of R2	p. 441
Unbounded Functions on Bounded Subsets	p. 443
Notes and Comments	p. 447
Exercises	p. 449
References	p. 463
List of Symbols and Abbreviations	p. 467
Index	p. 471
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