Advanced Calculus An Introduction to Classical Analysis

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ISBN: 9780486445489 | 0486445488
Cover: Paperback
Copyright: 1/30/2006

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A course in analysis that focuses on the functions of a real variable, this text is geared toward upper-level undergraduate students. It introduces the basic concepts in their simplest setting and illustrates its teachings with numerous examples, practical theorems, and coherent proofs. Starting with the structure of the system of real and complex numbers, it covers the convergence of sequences and series and explores the functions of a real variable and of several variables. Subsequent chapters offer a brief, self-contained introduction to vectors and the important differential invariants, the reversal of order in limiting processes, and Fourier series. 1955 ed.

The Number System

Groups

(3)

Fields

(1)

Integers

(3)

Rational Numbers

(2)

Properties of Rational Numbers

(3)

Real Numbers

(3)

Operations with Real Numbers

(2)

Complex Numbers

(3)

Complex Numbers as Vectors

(2)

Development of the Number System

(4)

Sequences and Series

Linear Point Sets

(1)

Cluster Points

(1)

Bounded Sets

(3)

Sequences

(1)

Limit of a Sequence

(5)

Operations with Limits

(2)

Fundamental Convergence Criterion

(1)

Monotone Sequences

(1)

The Number e

(2)

Nested Intervals

(2)

Infinite Series

(1)

Geometric Series

(1)

Positive Series

(1)

Telescopic Series

(3)

Comparison Tests

(1)

Cauchy's Condensation Test

(3)

Limit Form of Comparison Test

(1)

Cauchy's Root Test

(1)

d'Alembert's Ratio Test

(3)

Kummer-Jensen Tests

(2)

Gauss' Test

(1)

Absolute Convergence

(1)

Conditional Convergence

(1)

Alternating Series

(2)

Addition of Series

(1)

Rearrangement of Series

(3)

Multiplication of Series

(2)

Power Series

(2)

Binomial Series

(2)

Complex Sequences

(1)

Complex Series

(4)

Sequences and Series

(3)

Functions of a Real Variable

Functions

(2)

Limit of a Continuous Variable

(4)

Continuity at a Point

(2)

Continuity in an Interval

(3)

Bounds of a Continuous Function

(2)

Intermediate Values

(1)

Inverse Functions

(3)

Derivative

100

(4)

Increasing and Decreasing Functions

104

(1)

The Chain Rule

104

(1)

Derivative of an Inverse Function

105

(4)

Higher Derivatives

109

(3)

Rolle's Theorem

112

(1)

Theorem of Darboux

112

(1)

Mean-Value Theorem

113

(3)

The Iterative Solution of Equations

116

(2)

Second Mean-Value Theorem (Cauchy)

118

(1)

l'Hospital's Rule

119

(3)

The Form ∞/∞

122

(3)

Other-Indeterminate Forms

125

(1)

Approximating Polynomial

126

(1)

The Remainder

127

(3)

Taylor's Theorem

130

(2)

Exponential Series

132

(1)

Sine and Cosine Series

133

(1)

Even and Odd Functions

134

(1)

Logarithmic Series

135

(1)

Binomial Series

136

(2)

Extremes of f(x)

138

(5)

Summary: Functions of a Real Variable

143

(4)

Functions of Several Variables

Functions of Two Variables

147

(2)

Continuity at a Point

149

(2)

Continuity in a Region

151

(1)

Partial Derivatives

152

(1)

Total Differential

153

(2)

Differentiable Functions

155

(1)

Composite Functions

156

(3)

Functions of Three Variables

159

(2)

Homogeneous Functions

161

(1)

Higher Derivatives

162

(3)

Implicit Functions

165

(3)

One Equation

168

(2)

Two Equations

170

(4)

Three Equations

174

(1)

Coordinate Transformations

175

(4)

Jacobians

179

(1)

Functional Dependence

180

(4)

Mean-Value Theorem for f(x, y)

184

(1)

Taylor's Theorem for f(x, y)

185

(1)

Extremes of f(x, y)

186

(6)

Constrained Extremes

192

(2)

Lagrangian Multipliers

194

(4)

Summary: Functions of Several Variables

198

(4)

Vectors

202

(5)

Scalar Product

207

(2)

Vector Product

209

(2)

Box Product

211

(3)

Derivative of a Vector

214

(1)

Curves

215

(2)

Unit Tangent Vector

217

(3)

Frenet's Formulas

220

(4)

Curvature and Torsion

224

(3)

Surfaces

227

(3)

Directional Derivative

230

(5)

Gradient of a Vector

235

(1)

Invariants of a Dyadic

236

(1)

Divergence and Rotation

237

(2)

Reciprocal Bases

239

(1)

Curvilinear Coordinates

240

(5)

Vector Algebra and Calculus

245

(3)

The Definite Integral

The Riemann Integral

248

(2)

Condition for Integrability

250

(3)

Integrable Functions

253

(3)

Integration by Summation

256

(1)

Properties of the Integral

257

(2)

Formation of Integrable Functions

259

(1)

The Integral as a Function of Its Upper Limit

260

(2)

The Fundamental Theorem of the Integral Calculus

262

(4)

Mean Value Theorems for Integrals

266

(1)

Integration by Parts

267

(3)

Change of Variable

270

(5)

Algebraic Integrals

275

(2)

Duhamel's Theorem

277

(1)

Areas

278

(3)

Functions of Bounded Variation

281

(1)

Length of a Curve

282

(2)

Smooth Curves

284

(5)

Approximate Integration

289

(2)

Error in Simpson's Rule

291

(3)

The Integral as a Function of a Parameter

294

(1)

Differentiation of Integrals

294

(5)

Application to Differential Equations

299

(1)

Repeated Integrals

300

(1)

Integrals

301

(3)

Improper Integrals

Types of Improper Integrals

304

(2)

Evaluation of Improper Integrals

306

(2)

Analogy, with Series

308

(3)

Comparison Tests: Type I

311

(1)

Absolute Convergence: Type I

312

(1)

Limit Tests: Type I

312

(2)

Comparison Tests: Type II

314

(1)

Limit Tests: Type II

315

(3)

Conditional Convergence: Type I

318

(2)

Conditional Convergence: Type II

320

(1)

Combinations of Types I and II

321

(2)

Laplace Transform

323

(3)

Convergence of Improper Integrals

326

(2)

Line Integrals

328

(3)

Line Integrals Independent of Path

331

(3)

Field of Force

334

(2)

Irrotational Vectors

336

(5)

Area of a Sector

341

(4)

Multiple Integrals

Double Integral over a Rectangle

345

(1)

Condition for Integrability

346

(2)

Continuity of an Integral

348

(2)

Double Integral within a Curve

350

(1)

Double and Repeated Integrals

351

(4)

Green's Theorem in the Plane

355

(7)

Element of Area

362

(2)

Change of Variables in a Double Integral

364

(2)

Curves on a Surface

366

(3)

Area of a Surface

369

(4)

Surface Integral

373

(2)

Stokes' Theorem

375

(4)

Line Integrals in Space

379

(2)

Triple Integral over a Rectangular Prism

381

(1)

Element of Volume

382

(2)

Triple and Repeated Integrals

384

(3)

Divergence Theorem

387

(4)

Solenoidal Vectors

391

(5)

Line, Surface, and Volume Integrals

396

(3)

Uniform Convergence

Reversal of Order in Limiting Processes

399

(1)

Uniform Convergence of a Sequence

400

(2)

Continuity of the Limit Function

402

(3)

Integrals in a Sequence

405

(1)

Derivatives of a Sequence

406

(2)

Uniform Convergence of a Series

408

(4)

Continuity of the Sum

412

(1)

Integration of Series

412

(1)

Differentiation of Series

413

(3)

Power Series

416

(2)

Abel's Theorem

418

(4)

Consequences of Abel's Theorem

422

(4)

Uniform Convergence of Improper Integrals

426

(2)

M-Test for Integrals

428

(2)

Continuity of Improper Integrals

430

(1)

Integration of Improper Integrals

431

(3)

Differentiation of Improper Integrals

434

(2)

Gamma Function

436

(3)

Beta Function

439

(1)

Relation between Beta and Gamma Function

440

(4)

Uniform Convergence

444

(2)

Functions of a Complex Variable

Rational Functions of a Complex Variable

446

(2)

Functions of a Complex Variable

448

(1)

Analytic Functions

449

(4)

Exponential Functions

453

(1)

Sine and Cosine

454

(2)

Hyperbolic Functions

456

(1)

Trigonometric Relations

457

(1)

Logarithm

458

(2)

Conformal Mapping

460

(7)

Definite Integrals

467

(3)

Cauchy's Integral Theorem

470

(3)

Cauchy's Integral Formula

473

(1)

Complex Taylor Series

474

(2)

Cauchy's Inequality

476

(1)

Isolated Singularities

477

(5)

Laurent Series

482

(4)

Bernoulli Numbers

486

(4)

Reciprocal of a Function

490

(2)

Residues

492

(2)

Residue Theorem

494

(3)

Evaluation of Definite Integrals

497

(2)

Improper Real Integrals

499

(6)

Indented Contours

505

(3)

Properties of Analytic Functions

508

(3)

Fourier Series

Orthogonal Sets of Functions

511

(3)

Closed and Complete Orthonormal Sets

514

(1)

Fourier Series

515

(4)

Convergence Theorem

519

(2)

Convergence at Discontinuities

521

(5)

Resolution of cot πx into Partial Fractions

526

(1)

Approximation Theorems

527

(2)

Parseval's Theorem

529

(2)

Integration of Fourier Series

531

(3)

Uniform Convergence of Fourier Series

534

(2)

Gibbs' Phenomenon

536

(2)

Properties of Fourier Series

538

(2)

Appendix 1. Cluster Points

540

(2)

Appendix 2. Difference Equations

542

(3)

Appendix 3. The Difference Calculus

545

(4)

Appendix 4. Dimensional Checks

549

(2)

Comprehensive Test

551

(2)

Answers to Problems

553

(14)

Index

567

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Advanced Calculus An Introduction to Classical Analysis

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