Advanced Calculus for Mathematical Modeling in Engineering and Physics introduces the principles and methods of advanced calculus for mathematical modeling, through a balance of theory and application using a state space approach with elementary functional analysis. This framework facilitates a deeper understanding of the nature of mathematical models and of the behavior of their solutions. The work provides a variety of advanced calculus models for mathematical, physical science, and engineering audiences, with discussion of how calculus-based models and their discrete analogies are generated. This valuable textbook offers scientific computations driven by Octave/MATLAB script, in recognition of the rising importance of associated numerical models.
1. 1.5 Vector calculus with an introduction to Hilbert Spaces - p. 64 2. 1.6 Homogeneous linear models - p. 76 3. 1.7 Nonhomogeneous linear models - p. 83 4. 1.8 ∗∗Quantitative error analysis and numerical methods for linear models (optional) - p. 88 5. 1.9 Problems 1 - p. 100 6. Chapter 2. Ordinary differential equations and recurrence relations - p. 115 7. 2.1 Classical solutions to initial value problems - p. 118 8. 2.1.1 Existence and uniqueness of solutions - p. 118 9. First-order DE-based IVPs - p. 118 10. Theorems for nth order DE-based IVPs - p. 125 11. 2.1.2 Nonlinear DE-based IVPs - p. 129 12. Integrating factors to solve first-order DEs - p. 129 13. Substitution methods to solve first-order DEs - p. 131 14. Reducing the order of a second-order DE28 - p. 136 15. 2.1.3 Nonlinear recurrence relation-based IVPs - p. 138 16. 2.1.4 Modeling with differential equations (optional) - p. 146 17. Population dynamics as an example of iterative model development - p. 148 18. Modeling the spread of disease - p. 150 19. Compartment models - p. 151 20. Cooling and heating - p. 153 21. 2.1.5 Modeling with second-order Linear IVPs (optional) - p. 163 22. Applications of DEs with constant coefficients - p. 164 23. Vibration and decay in basic second-order DE models - p. 165 24. Nondimensionalization of a DE - p. 180 25. 2.1.6 Linear DE-based IVPs of general order - p. 183 26. Homogeneous higher order linear DEs - p. 183 27. Nonhomogeneous higher order linear equations - p. 189 28. Incorporating initial conditions (to solve initial value problems) - p. 191 29. 2.1.7 Linear recurrence relation-based IVPs - p. 192 30. First-order initial value problems - p. 192 31. Higher order homogeneous linear IVPs - p. 194 32. Nonhomogeneous IVPs - p. 203 33. 2.1.8 Problems 2.1 - p. 207 34. 2.2 Series solution methods - p. 219 35. 2.2.1 Power series method for analytic solutions - p. 220 36. Review of theorems for power series - p. 220 37. Power series solution at an ordinary point of a second order homogeneous linear DE - p. 222 38. Recurrence relations for series coefficients - p. 224 39. 2.2.1.4 Translation of series t0=0 - p. 228 40. Polynomial and tabulated solutions - p. 230 41. Nonhomogeneous problems - p. 234 42. 2.2.2 Series solutions at regular singular points: Frobenius' method - p. 235 43. 2.2.3 Homogeneous BVPs, Sturm-Liouville theory, and series for functions in L2[a,b] - p. 243 44. Regular Sturm–Liouville boundary value problems - p. 244 45. Higher order BVPs - p. 247 46. Representation of functions as series of Sturm–Liouville eigenfunctions - p. 248 47. Fourier sine and cosine series representation of functions in L2[0,π] - p. 250 48. Representation by other S-L eigenfunctions - p. 252 49. 2.2.4 Nonhomogeneous BVPs: The Fredholm alternative and Green's functions - p. 256 50. 2.2.5 Problems 2.2 - p. 265 51. 2.3 Transform methods - p. 270 52. 2.3.1 Basic Laplace transforms - p. 271 53. 2.3.2 Properties of the Laplace transform - p. 273 54. 2.3.3 Laplace transform method to solve DE-based IVPs - p. 277 55. 2.3.4 Piecewise continuous, impulsive, and periodic forcing - p. 281 56. 2.3.5 Z-transform method for recurrence relation-based IVPs - p. 291 57. 2.3.6 Problems 2.3 - p. 296 58. 2.4 Approximation of solutions to differential equations - p. 299 59. Chapter 3. Systems of differential equations and discrete analogies - p. 317 60. 3.1 Linear and nonlinear systems of DEs - p. 318 61. 3.2 Homogeneous linear systems - p. 325 62. 3.2.1 First-order systems of DEs with constant coefficients - p. 328 63. Solutions of real systems - p. 328 64. Solution of systems with A complex - p. 335 65. 3.2.2 Real second-order DE systems with constant coefficients - p. 335 66. 3.2.3 Matrix exponential methods for DE systems - p. 339 67. 3.2.4 Systems of homogeneous linear recurrence relations and numerical eigenvalue methods (optional) - p. 344 68. 3.2.5 Problems 3.2 - p. 351 69. 3.3 Nonhomogeneous linear systems - p. 355 70. 3.3.1 Undetermined coefficients for DE systems - p. 356 71. 3.3.2 Particular solution by variation of parameters or transform - p. 357 72. 3.3.3 Systems of nonhomogeneous linear recurrence relations (optional) - p. 360 73. 3.3.4 Problems 3.3 - p. 371 74. 3.4 Qualitative analysis of autonomous nonlinear systems of DEs - p. 372 75. 3.4.1 Phase space - p. 373 76. 3.4.2 Almost linear systems and linearization - p. 378 77. 3.4.3 Stability of isolated critical points - p. 389 78. 3.4.4 Structural stability and bifurcation - p. 403 79. 3.4.5 Problems 3.4 - p. 412 80. 3.5 Approximation of solutions - p. 416 81. Chapter 4. Integration for advanced calculus models - p. 425 82. 4.1 Integration in real Euclidean space - p. 425 83. 4.1.1 Riemann versus Lebesgue integration - p. 425 84. Coordinate transformations in multiple integrals and vector operators - p. 426 85. Theorems of Green, Stokes, and Gauss - p. 437 86. Models and evaluation of difficult integrals through theorems - p. 446 87. ∗∗Machine symbolic and numerical integration - p. 450 88. Problems 4.1 - p. 459 89. 4.2 Functions and constants defined as integrals - p. 466 90. 4.2.1 Integrals for applications - p. 468 91. Gamma function - p. 468 92. Fractional calculus - p. 469 93. Laplace transform of a power function with an irrational power - p. 471 94. Elliptic integrals - p. 472 95. Gaussian integral, error function, and relatives - p. 473 96. Integration of inverse functions - p. 475 97. Cardinal sine function and sine integral - p. 476 98. Fresnel integrals - p. 477 99. Riemann zeta function - p. 479 100. 4.2.2 Asymptotic expansions - p. 481 101. 4.2.3 Problems 4.2 - p. 483 102. 4.3 Integral transforms and the discrete fourier transform analogy - p. 488 103. 4.3.1 Fourier transform and related integral transforms - p. 488 104. Fourier transform - p. 488 105. Fourier sine and cosine transforms - p. 490 106. Representation of odd and even functions - p. 492 107. Inverse Fourier transform and Fourier integral representation - p. 495 108. State space for Fourier transforms and transforms for applications - p. 496 109. Related integral transforms and applications - p. 504 110. 4.3.2∗∗ Discrete Fourier transform (optional) - p. 505 111. 4.3.3 Problems 4.3 - p. 512 112. 4.4 Models through functional considerations - p. 513 113. 4.5∗ Continuous probability theory (optional) - p. 525 114. 4.5.1 Single variable theory - p. 526 115. Probability distribution of a single random variable - p. 526 116. Characteristic function of a distribution - p. 529 117. Normal distribution - p. 531 118. Confidence intervals - p. 537 119. Normal approximation to the binomial distribution - p. 539 120. 4.5.2∗ Multivariate probability theory (optional) - p. 541 121. 4.5.3 Problems 4.5 - p. 548 122. Chapter 5. Partial differential equations and applications - p. 553 123. 5.1 Boundary value problems - p. 554 124. 5.1.1 Representation of functions by fourier series or other series of orthogonal functions - p. 554 125. 5.1.2 Solution of BVPs via orthogonal series or integral transform - p. 566 126. Waves - p. 566 127. Diffusion - p. 569 128. Steady-state solutions - p. 571 129. PDEs in higher dimensions - p. 572 130. Solutions on semi-infinite regions - p. 579 131. 5.1.3 Problems 5.1 - p. 581 132. 5.2 Solution by the method of characteristics - p. 583 133. 5.2.1 Quasilinear first-order PDEs - p. 584 134. 5.2.2 Linear second-order PDEs - p. 591 135. Solution through substitution to get an ordinary DE - p. 591 136. Second-order PDEs with constant coefficients - p. 592 137. Classification of second-order linear PDEs - p. 596 138. Constant coefficients - p. 597 139. Hyperbolic PDEs - p. 599 140. Parabolic PDEs - p. 600 141. Elliptic PDEs - p. 601 142. 5.2.3 Problems 5.2 - p. 603 143. Chapter 6. Calculus of variations - p. 607 144. 6.1 Simplest problem and applications - p. 608 145. 6.1.1 Equations of the simplest problem - p. 608 146. 6.1.2 Models through the calculus of variations - p. 614 147. 6.1.3 Problems 6.1 - p. 621 148. 6.2 Problems in more variables - p. 624 149. Chapter 7. Calculus with complex numbers - p. 641 150. 7.1 Complex limit and derivative - p. 645 151. 7.2 Analytic continuation of familiar functions - p. 656 152. 7.3 Mapping with complex functions - p. 665 153. 7.4 Complex integration - p. 676 154. 7.5 Sequences and series - p. 691 155. 7.6 Power series and Laurent series - p. 697 156. 7.7 Residue theorem, contour integration, and improper integrals - p. 707 157. 7.8 Inverse Laplace transforms - p. 716 158. 7.9 Problems 7 - p. 719 159. Chapter 8. Introduction to tensors - p. 727 160. 8.1 Cartesian tensors - p. 728 161. 8.2 Dual spaces - p. 736 162. 8.3 Tensors - p. 743 163. 8.4 Tensor fields on real manifolds - p. 754 164. 8.5 Tensor algebra - p. 760 165. 8.6 Covariant differentiation - p. 764 166. 8.6.1 Covariant differentiation of a real tensor field - p. 764 167. Covariant differentiation of a scalar field - p. 764 168. Covariant differential of a vector field - p. 765 169. Second derivatives, curvature, and torsion - p. 770 170. 8.6.2 Euclidean space in curvilinear coordinates - p. 772 171. 8.6.3 Isotropic tensors and relative tensors - p. 774 172. Determinants - p. 775 173. 8.6.4 Derivative and arc length along a curve - p. 779 174. 8.7 Differential forms and integration - p. 782 175. 8.8 Problems 8 - p. 795 176. Appendices - p. 801 177. A.1 Appendix A: Second-order linear differential equations - p. 801 178. A.1.1 General solution of homogeneous DEs - p. 801 179. A.1.2 General solution of nonhomogeneous DEs - p. 803 180. B.1.1 Review of eigenvalues and eigenvectors of square matrices - p. 806 181. B.1.2 Diagonalization of a square Matrix - p. 806 182. B.1.3 Generalized Eigenvectors - p. 807 183. B.1.4 Special matrices for diagonalization of normal and symmetric matrices - p. 809 184. B.1.5 Unitary and orthogonal diagonalization - p. 811 185. C.1.1 Matrix representation of a linear model Lu=b - p. 812 186. C.1.2 Jordan canonical form, QR factorization, and SVD of matrices - p. 816 187. C.1.3 Least squares solution of overdetermined systems Au=b - p. 823 188. Solution by the normal equations - p. 824 189. Solution by QR factorization - p. 824 190. Solution by SVD - p. 826 191. Pseudoinverse of a matrix and generalized least squares problems - p. 828 192. D.1.1 Line integrals - p. 830 193. Scalar line integrals - p. 830 194. Vector line integrals - p. 832 195. Line integrals in conservative vector fields - p. 836 196. D.1.2 Double integrals - p. 838 197. D.1.3 Vector Surface Integrals - p. 850 198. Index - p. 853
Advanced Calculus for Mathematical Modeling in Engineering and Physics introduces the principles and methods of advanced calculus for mathematical modeling, through a balance of theory and application using a state space approach with elementary functional analysis. This framework facilitates a deeper understanding of the nature of mathematical models and of the behavior of their solutions. The work provides a variety of advanced calculus models for mathematical, physical science, and engineering audiences, with discussion of how calculus-based models and their discrete analogies are generated. This valuable textbook offers scientific computations driven by Octave/MATLAB script, in recognition of the rising importance of associated numerical models.
1. 1.5 Vector calculus with an introduction to Hilbert Spaces - p. 64 2. 1.6 Homogeneous linear models - p. 76 3. 1.7 Nonhomogeneous linear models - p. 83 4. 1.8 ∗∗Quantitative error analysis and numerical methods for linear models (optional) - p. 88 5. 1.9 Problems 1 - p. 100 6. Chapter 2. Ordinary differential equations and recurrence relations - p. 115 7. 2.1 Classical solutions to initial value problems - p. 118 8. 2.1.1 Existence and uniqueness of solutions - p. 118 9. First-order DE-based IVPs - p. 118 10. Theorems for nth order DE-based IVPs - p. 125 11. 2.1.2 Nonlinear DE-based IVPs - p. 129 12. Integrating factors to solve first-order DEs - p. 129 13. Substitution methods to solve first-order DEs - p. 131 14. Reducing the order of a second-order DE28 - p. 136 15. 2.1.3 Nonlinear recurrence relation-based IVPs - p. 138 16. 2.1.4 Modeling with differential equations (optional) - p. 146 17. Population dynamics as an example of iterative model development - p. 148 18. Modeling the spread of disease - p. 150 19. Compartment models - p. 151 20. Cooling and heating - p. 153 21. 2.1.5 Modeling with second-order Linear IVPs (optional) - p. 163 22. Applications of DEs with constant coefficients - p. 164 23. Vibration and decay in basic second-order DE models - p. 165 24. Nondimensionalization of a DE - p. 180 25. 2.1.6 Linear DE-based IVPs of general order - p. 183 26. Homogeneous higher order linear DEs - p. 183 27. Nonhomogeneous higher order linear equations - p. 189 28. Incorporating initial conditions (to solve initial value problems) - p. 191 29. 2.1.7 Linear recurrence relation-based IVPs - p. 192 30. First-order initial value problems - p. 192 31. Higher order homogeneous linear IVPs - p. 194 32. Nonhomogeneous IVPs - p. 203 33. 2.1.8 Problems 2.1 - p. 207 34. 2.2 Series solution methods - p. 219 35. 2.2.1 Power series method for analytic solutions - p. 220 36. Review of theorems for power series - p. 220 37. Power series solution at an ordinary point of a second order homogeneous linear DE - p. 222 38. Recurrence relations for series coefficients - p. 224 39. 2.2.1.4 Translation of series t0=0 - p. 228 40. Polynomial and tabulated solutions - p. 230 41. Nonhomogeneous problems - p. 234 42. 2.2.2 Series solutions at regular singular points: Frobenius' method - p. 235 43. 2.2.3 Homogeneous BVPs, Sturm-Liouville theory, and series for functions in L2[a,b] - p. 243 44. Regular Sturm–Liouville boundary value problems - p. 244 45. Higher order BVPs - p. 247 46. Representation of functions as series of Sturm–Liouville eigenfunctions - p. 248 47. Fourier sine and cosine series representation of functions in L2[0,π] - p. 250 48. Representation by other S-L eigenfunctions - p. 252 49. 2.2.4 Nonhomogeneous BVPs: The Fredholm alternative and Green's functions - p. 256 50. 2.2.5 Problems 2.2 - p. 265 51. 2.3 Transform methods - p. 270 52. 2.3.1 Basic Laplace transforms - p. 271 53. 2.3.2 Properties of the Laplace transform - p. 273 54. 2.3.3 Laplace transform method to solve DE-based IVPs - p. 277 55. 2.3.4 Piecewise continuous, impulsive, and periodic forcing - p. 281 56. 2.3.5 Z-transform method for recurrence relation-based IVPs - p. 291 57. 2.3.6 Problems 2.3 - p. 296 58. 2.4 Approximation of solutions to differential equations - p. 299 59. Chapter 3. Systems of differential equations and discrete analogies - p. 317 60. 3.1 Linear and nonlinear systems of DEs - p. 318 61. 3.2 Homogeneous linear systems - p. 325 62. 3.2.1 First-order systems of DEs with constant coefficients - p. 328 63. Solutions of real systems - p. 328 64. Solution of systems with A complex - p. 335 65. 3.2.2 Real second-order DE systems with constant coefficients - p. 335 66. 3.2.3 Matrix exponential methods for DE systems - p. 339 67. 3.2.4 Systems of homogeneous linear recurrence relations and numerical eigenvalue methods (optional) - p. 344 68. 3.2.5 Problems 3.2 - p. 351 69. 3.3 Nonhomogeneous linear systems - p. 355 70. 3.3.1 Undetermined coefficients for DE systems - p. 356 71. 3.3.2 Particular solution by variation of parameters or transform - p. 357 72. 3.3.3 Systems of nonhomogeneous linear recurrence relations (optional) - p. 360 73. 3.3.4 Problems 3.3 - p. 371 74. 3.4 Qualitative analysis of autonomous nonlinear systems of DEs - p. 372 75. 3.4.1 Phase space - p. 373 76. 3.4.2 Almost linear systems and linearization - p. 378 77. 3.4.3 Stability of isolated critical points - p. 389 78. 3.4.4 Structural stability and bifurcation - p. 403 79. 3.4.5 Problems 3.4 - p. 412 80. 3.5 Approximation of solutions - p. 416 81. Chapter 4. Integration for advanced calculus models - p. 425 82. 4.1 Integration in real Euclidean space - p. 425 83. 4.1.1 Riemann versus Lebesgue integration - p. 425 84. Coordinate transformations in multiple integrals and vector operators - p. 426 85. Theorems of Green, Stokes, and Gauss - p. 437 86. Models and evaluation of difficult integrals through theorems - p. 446 87. ∗∗Machine symbolic and numerical integration - p. 450 88. Problems 4.1 - p. 459 89. 4.2 Functions and constants defined as integrals - p. 466 90. 4.2.1 Integrals for applications - p. 468 91. Gamma function - p. 468 92. Fractional calculus - p. 469 93. Laplace transform of a power function with an irrational power - p. 471 94. Elliptic integrals - p. 472 95. Gaussian integral, error function, and relatives - p. 473 96. Integration of inverse functions - p. 475 97. Cardinal sine function and sine integral - p. 476 98. Fresnel integrals - p. 477 99. Riemann zeta function - p. 479 100. 4.2.2 Asymptotic expansions - p. 481 101. 4.2.3 Problems 4.2 - p. 483 102. 4.3 Integral transforms and the discrete fourier transform analogy - p. 488 103. 4.3.1 Fourier transform and related integral transforms - p. 488 104. Fourier transform - p. 488 105. Fourier sine and cosine transforms - p. 490 106. Representation of odd and even functions - p. 492 107. Inverse Fourier transform and Fourier integral representation - p. 495 108. State space for Fourier transforms and transforms for applications - p. 496 109. Related integral transforms and applications - p. 504 110. 4.3.2∗∗ Discrete Fourier transform (optional) - p. 505 111. 4.3.3 Problems 4.3 - p. 512 112. 4.4 Models through functional considerations - p. 513 113. 4.5∗ Continuous probability theory (optional) - p. 525 114. 4.5.1 Single variable theory - p. 526 115. Probability distribution of a single random variable - p. 526 116. Characteristic function of a distribution - p. 529 117. Normal distribution - p. 531 118. Confidence intervals - p. 537 119. Normal approximation to the binomial distribution - p. 539 120. 4.5.2∗ Multivariate probability theory (optional) - p. 541 121. 4.5.3 Problems 4.5 - p. 548 122. Chapter 5. Partial differential equations and applications - p. 553 123. 5.1 Boundary value problems - p. 554 124. 5.1.1 Representation of functions by fourier series or other series of orthogonal functions - p. 554 125. 5.1.2 Solution of BVPs via orthogonal series or integral transform - p. 566 126. Waves - p. 566 127. Diffusion - p. 569 128. Steady-state solutions - p. 571 129. PDEs in higher dimensions - p. 572 130. Solutions on semi-infinite regions - p. 579 131. 5.1.3 Problems 5.1 - p. 581 132. 5.2 Solution by the method of characteristics - p. 583 133. 5.2.1 Quasilinear first-order PDEs - p. 584 134. 5.2.2 Linear second-order PDEs - p. 591 135. Solution through substitution to get an ordinary DE - p. 591 136. Second-order PDEs with constant coefficients - p. 592 137. Classification of second-order linear PDEs - p. 596 138. Constant coefficients - p. 597 139. Hyperbolic PDEs - p. 599 140. Parabolic PDEs - p. 600 141. Elliptic PDEs - p. 601 142. 5.2.3 Problems 5.2 - p. 603 143. Chapter 6. Calculus of variations - p. 607 144. 6.1 Simplest problem and applications - p. 608 145. 6.1.1 Equations of the simplest problem - p. 608 146. 6.1.2 Models through the calculus of variations - p. 614 147. 6.1.3 Problems 6.1 - p. 621 148. 6.2 Problems in more variables - p. 624 149. Chapter 7. Calculus with complex numbers - p. 641 150. 7.1 Complex limit and derivative - p. 645 151. 7.2 Analytic continuation of familiar functions - p. 656 152. 7.3 Mapping with complex functions - p. 665 153. 7.4 Complex integration - p. 676 154. 7.5 Sequences and series - p. 691 155. 7.6 Power series and Laurent series - p. 697 156. 7.7 Residue theorem, contour integration, and improper integrals - p. 707 157. 7.8 Inverse Laplace transforms - p. 716 158. 7.9 Problems 7 - p. 719 159. Chapter 8. Introduction to tensors - p. 727 160. 8.1 Cartesian tensors - p. 728 161. 8.2 Dual spaces - p. 736 162. 8.3 Tensors - p. 743 163. 8.4 Tensor fields on real manifolds - p. 754 164. 8.5 Tensor algebra - p. 760 165. 8.6 Covariant differentiation - p. 764 166. 8.6.1 Covariant differentiation of a real tensor field - p. 764 167. Covariant differentiation of a scalar field - p. 764 168. Covariant differential of a vector field - p. 765 169. Second derivatives, curvature, and torsion - p. 770 170. 8.6.2 Euclidean space in curvilinear coordinates - p. 772 171. 8.6.3 Isotropic tensors and relative tensors - p. 774 172. Determinants - p. 775 173. 8.6.4 Derivative and arc length along a curve - p. 779 174. 8.7 Differential forms and integration - p. 782 175. 8.8 Problems 8 - p. 795 176. Appendices - p. 801 177. A.1 Appendix A: Second-order linear differential equations - p. 801 178. A.1.1 General solution of homogeneous DEs - p. 801 179. A.1.2 General solution of nonhomogeneous DEs - p. 803 180. B.1.1 Review of eigenvalues and eigenvectors of square matrices - p. 806 181. B.1.2 Diagonalization of a square Matrix - p. 806 182. B.1.3 Generalized Eigenvectors - p. 807 183. B.1.4 Special matrices for diagonalization of normal and symmetric matrices - p. 809 184. B.1.5 Unitary and orthogonal diagonalization - p. 811 185. C.1.1 Matrix representation of a linear model Lu=b - p. 812 186. C.1.2 Jordan canonical form, QR factorization, and SVD of matrices - p. 816 187. C.1.3 Least squares solution of overdetermined systems Au=b - p. 823 188. Solution by the normal equations - p. 824 189. Solution by QR factorization - p. 824 190. Solution by SVD - p. 826 191. Pseudoinverse of a matrix and generalized least squares problems - p. 828 192. D.1.1 Line integrals - p. 830 193. Scalar line integrals - p. 830 194. Vector line integrals - p. 832 195. Line integrals in conservative vector fields - p. 836 196. D.1.2 Double integrals - p. 838 197. D.1.3 Vector Surface Integrals - p. 850 198. Index - p. 853
