Page 14 - Advanced engineering mathematics
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viii Contents
CHAPTER 12 Vector Integral Calculus 367
12.1 Line Integrals 367
12.1.1 Line Integral With Respect to Arc Length 372
12.2 Green’s Theorem 374
12.3 An Extension of Green’s Theorem 376
12.4 Independence of Path and Potential Theory 380
12.5 Surface Integrals 388
12.5.1 Normal Vector to a Surface 389
12.5.2 Tangent Plane to a Surface 392
12.5.3 Piecewise Smooth Surfaces 392
12.5.4 Surface Integrals 393
12.6 Applications of Surface Integrals 395
12.6.1 Surface Area 395
12.6.2 Mass and Center of Mass of a Shell 395
12.6.3 Flux of a Fluid Across a Surface 397
12.7 Lifting Green’s Theorem to R 3 399
12.8 The Divergence Theorem of Gauss 402
12.8.1 Archimedes’s Principle 404
12.8.2 The Heat Equation 405
12.9 Stokes’s Theorem 408
12.9.1 Potential Theory in 3-Space 410
12.9.2 Maxwell’s Equations 411
12.10 Curvilinear Coordinates 414
PART 4 Fourier Analysis, Special Functions,
and Eigenfunction Expansions 425
CHAPTER 13 Fourier Series 427
13.1 Why Fourier Series? 427
13.2 The Fourier Series of a Function 429
13.2.1 Even and Odd Functions 436
13.2.2 The Gibbs Phenomenon 438
13.3 Sine and Cosine Series 441
13.3.1 Cosine Series 441
13.3.2 Sine Series 443
13.4 Integration and Differentiation of Fourier Series 445
13.5 Phase Angle Form 452
13.6 Complex Fourier Series 457
13.7 Filtering of Signals 461
CHAPTER 14 The Fourier Integral and Transforms 465
14.1 The Fourier Integral 465
14.2 Fourier Cosine and Sine Integrals 468
14.3 The Fourier Transform 470
14.3.1 Filtering and the Dirac Delta Function 481
14.3.2 The Windowed Fourier Transform 483
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October 15, 2010 17:43 THM/NEIL Page-viii 27410_00_fm_pi-xiv
