| Preface |
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v | |
| How this book is organized |
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viii | |
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1 | (433) |
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Vectors and Vector Calculus |
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2 | (97) |
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Vectors and Vector Spaces |
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2 | (28) |
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2 | (4) |
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Basis Vectors and Components of a Vector |
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6 | (4) |
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Three-Dimensional Cartesian Coordinate System and R3 |
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10 | (2) |
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12 | (6) |
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18 | (2) |
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20 | (4) |
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24 | (6) |
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30 | (15) |
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Vector Calculus: Gradient |
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30 | (1) |
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Geometrical Interpretation of a Gradient |
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31 | (4) |
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Vector Calculus: Divergence |
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35 | (2) |
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37 | (2) |
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Vector Calculus: Successive Applications of the Gradient Operator |
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39 | (6) |
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45 | (21) |
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Integral Theorems of Vector Calculus |
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66 | (33) |
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Vector Calculus: Line Integral |
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66 | (3) |
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Vector Calculus: Surface Integral |
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69 | (4) |
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Vector Calculus: Divergence Theorem |
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73 | (4) |
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Vector Calculus: Gauss' Law |
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77 | (3) |
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Vector Calculus: Stokes's Theorem |
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80 | (6) |
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Conservative Force and Potential Theory |
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86 | (5) |
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91 | (8) |
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99 | (86) |
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99 | (30) |
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The Vector Space of the Column Vectors. The Inner Product |
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103 | (3) |
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Writing Vectors in terms of an Orthonormal Basis |
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106 | (2) |
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Change of the Coordinate System. The Rotation Matrix |
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108 | (6) |
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Determinant and Matrix Inversion |
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114 | (11) |
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125 | (4) |
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129 | (25) |
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Some Applications of the Eigenvalue Problem, Diagonalization |
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135 | (4) |
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Orthogonal and Unitary Matrices |
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139 | (4) |
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Hermitian and Symmetric Matrices |
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143 | (11) |
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Rotation Transformations and Special Orthogonal Matrices |
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154 | (12) |
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Group Theory and Rotations |
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166 | (19) |
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167 | (5) |
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The Infinitesimal Rotations |
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172 | (5) |
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177 | (8) |
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185 | (42) |
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First Order Differential Equations |
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185 | (11) |
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186 | (1) |
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Exact First Order Differential Equations |
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187 | (3) |
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Linear First Order Differential Equations |
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190 | (6) |
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Second-Order Differential Equations |
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196 | (31) |
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Second Order Homogeneous Linear Differential Equation with Constant Coefficients |
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196 | (9) |
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Wronskian Representation of the Second-order Differential Equation |
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205 | (2) |
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Reduction of Order or How to Obtain a Missing Second Solution |
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207 | (1) |
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The Non-Homogeneous Equations |
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208 | (2) |
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Method of Undetermined Coefficients |
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210 | (1) |
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General Method for Particular Solutions |
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211 | (3) |
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Complex Method for the Exponential Source Term |
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214 | (1) |
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Complex Method for the Forced Oscillations |
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215 | (6) |
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Complex Method for Electric Circuits |
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221 | (6) |
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Series Solutions of Differential Equations |
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227 | (50) |
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Introduction to the Power Series Method |
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227 | (7) |
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Power Series Method, a Warm up Example |
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232 | (2) |
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Power Series Method, Expansion around a Regular Point |
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234 | (11) |
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Legendre's Differential Equation |
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235 | (10) |
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Frobenius' Method, Expansion around a Singular Point |
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245 | (21) |
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Bessel's Differential Equation |
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250 | (8) |
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Properties of Bessel Functions |
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258 | (1) |
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Reduction to Bessel's Equation |
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259 | (7) |
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Method of Separation of Variables; Helmholtz's and Laplace's Equations |
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266 | (11) |
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Separation in Spherical Coordinates |
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268 | (5) |
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Separation in Cylindrical Coordinates |
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273 | (4) |
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Special Functions and the Generalized Fourier Series |
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277 | (50) |
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Sturm-Liouville Theory and the Orthogonal Functions Expansion |
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277 | (19) |
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Introduction, Vibrations of the String |
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277 | (3) |
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The Sturm-Liouville Eigenvalue Problem |
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280 | (6) |
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286 | (3) |
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289 | (3) |
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292 | (4) |
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Harmonic Oscillator Equation, Periodic Sturm-Liouville Problem and Fourier Series |
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296 | (17) |
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Complex Form of Fourier Series |
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298 | (1) |
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Mean-Square Convergence of Fourier Series |
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299 | (2) |
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301 | (3) |
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Periodic Functions. Point-wise Convergence of Fourier Series |
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304 | (3) |
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307 | (6) |
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Dirac Delta Function and Fourier Integral Transform |
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313 | (14) |
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313 | (4) |
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Fourier Integral Transform |
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317 | (10) |
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Linear Systems of Differential Equations |
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327 | (44) |
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Homogeneous Systems of Differential Equations |
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327 | (17) |
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A Diagonalizable Coefficient Matrix |
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332 | (4) |
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336 | (8) |
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The Non-Homogeneous Linear Differential Equations |
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344 | (3) |
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Stability of Linear Systems |
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347 | (24) |
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Nonlinear Differential Equations |
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371 | (63) |
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One-Dimensional Nonlinear Differential Equations |
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371 | (7) |
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Two-Dimensional Nonlinear Differential Equations |
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378 | (24) |
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386 | (8) |
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Two-Dimensional Population Models |
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394 | (8) |
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Three-Dimensional Models; The Lorenz equations |
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402 | (5) |
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KdV Equation and Soliton Waves |
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407 | (27) |
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Darboux Transformations and One-Soliton Solution |
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415 | (4) |
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Crum Transformations and the Multi-Soliton Solutions |
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419 | (15) |
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434 | (230) |
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435 | (1) |
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Vectors and Vector Calculus |
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436 | (40) |
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436 | (9) |
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Vector Definition and Basic Operations with Vectors |
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437 | (3) |
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The Action of Matrices on Vectors |
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440 | (3) |
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Abstract Vector Manipulation |
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443 | (2) |
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Vector Differential Operators |
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445 | (9) |
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445 | (1) |
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Directional Derivative, and the Geometrical Interpretation of the Gradient |
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446 | (4) |
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Changing the Coordinate System |
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450 | (1) |
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Manipulating Vector Fields |
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451 | (1) |
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451 | (1) |
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452 | (1) |
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Successive Applications of Operator |
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453 | (1) |
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454 | (9) |
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456 | (2) |
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458 | (3) |
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461 | (2) |
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Scalar and Vector Potentials |
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463 | (2) |
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463 | (1) |
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464 | (1) |
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465 | (8) |
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466 | (2) |
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468 | (1) |
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Calculating the Flux of a Vector Field |
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468 | (2) |
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470 | (1) |
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470 | (2) |
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472 | (1) |
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473 | (3) |
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476 | (29) |
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476 | (1) |
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477 | (5) |
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477 | (1) |
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478 | (1) |
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479 | (1) |
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479 | (1) |
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479 | (1) |
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The Determinant of a Matrix |
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480 | (2) |
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Eigenvectors and Eigenvalues |
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482 | (3) |
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Diagonalization of Matrices |
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483 | (2) |
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485 | (6) |
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485 | (2) |
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487 | (2) |
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489 | (2) |
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Defining Matrices in Maple: General Properties |
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491 | (6) |
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495 | (2) |
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Matrix Exponential and the Group of Rotations |
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497 | (4) |
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501 | (4) |
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505 | (34) |
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First Order Differential Equations |
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505 | (13) |
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505 | (3) |
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Exact First Order Differential Equations |
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508 | (2) |
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Linear First Order Differential Equations |
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510 | (5) |
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Vector Fields Associated to First Order Differential Equations |
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515 | (3) |
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Compact Form of Displaying Differential Equations |
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518 | (3) |
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Second Order Differential Equations |
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521 | (11) |
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Second Order Homogeneous Linear Differential Equations with Constant Coefficients |
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521 | (7) |
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528 | (1) |
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Non-Homogeneous Second Order Linear Differential Equations |
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529 | (1) |
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General Method for Particular Solutions |
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530 | (2) |
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Numerical Solutions of Differential Equations |
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532 | (5) |
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537 | (2) |
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Power Series Solutions of Differential Equations |
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539 | (28) |
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539 | (4) |
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539 | (1) |
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540 | (1) |
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Radius of Convergence of a Series |
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541 | (1) |
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542 | (1) |
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Power Series Method: a Step-by-Step Example |
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543 | (1) |
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Power Series Method, Expansion Around the Regular Point |
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544 | (13) |
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Legendre's Differential Equation; Legendre's Polynomials |
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546 | (2) |
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548 | (3) |
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551 | (3) |
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554 | (3) |
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The Maple Package powseries |
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557 | (2) |
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Expansion Around the Singular Point: Frobenius Method |
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559 | (7) |
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Bessel's Differential Equation |
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562 | (4) |
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566 | (1) |
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Special Functions and Generalized Fourier Series |
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567 | (39) |
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The Sturm-Liouville Theory |
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567 | (25) |
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567 | (2) |
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Inner Product and Gram-Schmidt Orthogonalization of Functions |
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569 | (4) |
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573 | (5) |
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578 | (7) |
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585 | (7) |
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Periodic Sturm-Liouville Problem and the Fourier Series |
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592 | (7) |
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Trigonometric Form of the Fourier Series |
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592 | (6) |
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Complex Form of the Fourier Series |
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598 | (1) |
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Dirac Delta Function and the Fourier Transform |
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599 | (6) |
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599 | (3) |
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Fourier Integral Transform |
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602 | (3) |
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605 | (1) |
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Linear Systems of Differential Equations |
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606 | (23) |
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Homogeneous Systems of Differential Equations |
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607 | (4) |
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Non-Homogeneous Systems of Differential Equations |
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611 | (2) |
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Stability of Linear Systems |
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613 | (15) |
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614 | (3) |
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617 | (2) |
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Unstable Saddle (Hyperbolic) Point |
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619 | (1) |
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620 | (2) |
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622 | (1) |
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623 | (1) |
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624 | (2) |
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626 | (2) |
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628 | (1) |
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Nonlinear Differential Equations |
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629 | (35) |
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One-Dimensional Nonlinear Differential Equations |
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629 | (10) |
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Two-Dimensional Nonlinear Differential Equations |
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639 | (8) |
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Three-Dimensional Models; The Lorenz Equations |
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647 | (5) |
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KdV Equation and Soliton Waves |
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652 | (11) |
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654 | (2) |
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656 | (7) |
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663 | (1) |
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A. Complex Variables and Functions |
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664 | (12) |
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664 | (6) |
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Functions of Complex Variables |
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670 | (6) |
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676 | (17) |
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676 | (1) |
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677 | (1) |
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Manipulating Symbolic Expressions |
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677 | (2) |
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679 | (3) |
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682 | (1) |
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More on Expression Manipulation |
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683 | (2) |
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Solving Equations: The ``Exact'' Way |
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685 | (1) |
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Solving Equations: Numerical Solutions |
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686 | (2) |
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688 | (1) |
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689 | (3) |
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692 | (1) |
| Index for Part I |
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693 | (5) |
| Index for Part II |
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698 | |
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