1. Infinite Series, Power Series Page 1 1. The geometric series Page 1 2. Definitions and notation Page 3 3. Applications of series Page 5 4. Convergent and divergent series Page 5 5. Testing series for convergence; the preliminary test Page 7 6. Tests for convergence of series of positive terms; absolute convergence Page 8 7. Alternating series Page 15 8. Conditionally convergent series Page 16 9. Useful facts about series Page 17 10. Power series; interval of convergence Page 18 11. Theorems about power series Page 21 12. Expanding functions in power series Page 22 13. Techniques for obtaining power series expansions Page 24 14. Questions of convergence and accuracy in computation Page 29 15. Some uses of series Page 33 16. Miscellaneous problems Page 41 2. Complex Numbers Page 43 1. Introduction Page 43 2. Real and imaginary parts of a complex number Page 44 3. The complex plane Page 45 4. Terminology and notation Page 46 5. Complex algebra Page 48 6. Complex infinite series Page 54 7. Complex power series; circle of convergence Page 56 8. Elementary functions of complex numbers Page 58 9. Euler's formula Page 60 10. Powers and roots of complex numbers Page 63 11. The exponential and trigonometric functions Page 66 12. Hyperbolic functions Page 69 13. Logarithms Page 71 14. Complex roots and powers Page 72 15. Inverse trigonometric and hyperbolic functions Page 74 16. Some applications Page 76 17. Miscellaneous problems Page 79 3. Linear Equations; Vectors, Matrices, and Determinants Page 81 1. Introduction Page 81 2. Sets of linear equations, row reduction Page 82 3. Determinants; Cramer's rule Page 87 4. Vectors Page 95 5. Lines and planes Page 105 6. Matrix operations Page 113 7. Linear combinations, linear functions, linear operators Page 127 8. General theory of sets of linear equations Page 130 9. Special matrices Page 139 10. Miscellaneous problems Page 142 4. Partial Differentiation Page 145 1. Introduction and notation Page 145 2. Power series in two variables Page 148 3. Total differentials Page 150 4. Approximate calculations using differentials Page 154 5. Chain rule or differentiating a function of a function Page 156 6. Implicit differentiation Page 159 7. More chain rule Page 161 8. Application of partial differentiation to maximum and minimum problems Page 169 9. Maximum and minimum problems with constraints; Lagrange multipliers Page 172 10. Endpoint or boundary point problems Page 181 11. Change of variables Page 186 12. Differentiation of integrals; Leibniz' rule Page 192 13. Miscellaneous problems Page 197 5. Multiple Integrals; Applications of Integration Page 201 1. Introduction Page 201 2. Double and triple integrals Page 201 3. Applications of integration ; single and multiple integrals Page 208 4. Change of variables in integralS; Jacobians Page 217 5. Surface integrals Page 228 6. Miscellaneous problems Page 231 6. Vector Analysis Page 235 1. Introduction Page 235 2. Applications of vector multiplication Page 235 3. Triple products Page 237 4. Differentiation of vectors Page 244 5. Fields Page 248 6. Directional derivative; gradient Page 249 7. Some other expressions involving V Page 254 8. Line integrals Page 257 9. Green's theorem in the plane Page 266 10. The divergence and the divergence theorem Page 271 11. The curl and Stokes' theorem Page 281 12. Miscellaneous problems Page 293 7. Fourier Series Page 297 1. Introduction Page 297 2. Simple harmonic motion and wave motion; periodic functions Page 297 3. Applications of Fourier series Page 302 4. Average value of a function Page 304 5. Fourier coefficients Page 307 6. Dirichlet conditions Page 313 7. Complex form of Fourier series Page 315 8. Other intervals Page 317 9. Even and odd functions Page 321 10. An application to sound Page 328 11. Parseval's theorem Page 331 12. Miscellaneous problems Page 334 8. Ordinary Differential Equations Page 337 1. Introduction Page 337 2. Separable equations Page 341 3. Linear first-order equations Page 346 4. Other methods for first order equations Page 350 5. Second-order linear equations with constant coefficients and zero right-hand side Page 352 6. Second-order linear equations with constant coefficients and right-hand side not zero Page 361 7. Other second-order equations Page 374 8. Miscellaneous problems Page 379 9. Calculus of Variations Page 383 1. Introduction Page 383 2. The Euler equation Page 386 3. Using the Euler equation Page 389 4. The brachistochrone problem; cycloids Page 393 5. Several dependent variables; Lagrange's equations Page 396 6. Isoperimetric problems Page 401 7. Variational notation Page 403 8. Miscellaneous problems Page 404 10. Coordinate Transformations; Tensor Analysis Page 407 1. Introduction Page 407 2. Linear transformations Page 409 3. Orthogonal transformations Page 410 4. Eigenvalues and eigenvectors; diagonalizing matrices Page 413 5. Applications of diagonalization Page 420 6. Curvilinear coordinates Page 426 7. Scale factors and basis vectors for orthogonal systems Page 428 8. General curvilinear coordinates Page 429 9. Vector operators in orthogonal curvilinear coordinates Page 431 10. Tensor analysis—introduction Page 435 11. Cartesian tensors Page 437 12. Uses of tensors; dyadics Page 441 13. General coordinate systems Page 447 14. Vector operations in tensor notation Page 452 15. Miscellaneous problems Page 453 11. Gamma, Beta, and Error Functions; Asymptotic Series; Stirling's Formula; Elliptic Integrals and Functions Page 457 1. Introduction Page 457 2. The factorial function Page 457 3. Definition of the gamma function; recursion relation Page 458 4. The gamma function of negative numbers Page 460 5. Some important formulas involving gamma functions Page 461 6. Beta functions Page 462 7. The relation between the beta and gamma functions Page 463 8. The simple pendulum Page 465 9. The error function Page 467 10. Asymptotic series Page 469 11. Stirling's formula Page 472 12. Elliptic integrals and functions Page 474 13. Miscellaneous problems Page 481 12. Series Solutions of Differential Equations; Legendre Polynomials; Bessel Functions; Sets of Orthogonal Functions Page 483 1. Introduction Page 483 2. Legendre's equation Page 485 3. Leibniz' rule for differentiating products Page 488 4. Rodrigues' formula Page 489 5. Generating function for Legendre polynomials Page 490 6. Complete sets of orthogonal functions Page 496 7. Orthogonality of the Legendre polynomials Page 499 8. Normalization of the Legendre polynomials Page 500 9. Legendre series Page 502 10. The associated Legendre functions Page 504 11. Generalized power series or the method of Frobenius Page 506 12. Bessel's equation Page 509 13. The second solution of Bessel's equation Page 512 14. Tables, graphs, and zeros of Bessel functions Page 514 15. Recursion relations Page 514 16. A general differential equation having Bessel functions as solutions Page 516 17. Other kinds of Bessel functions Page 517 18. The lengthening pendulum Page 519 19. Orthogonality of Bessel functions Page 522 20. Approximate formulas for Bessel functions Page 525 21. Some general comments about series solutions Page 526 22. Hermite functions; Laguerre functions; ladder operators Page 530 23. Miscellaneous problems Page 537 13. Partial Differential Equations Page 541 1. Introduction Page 541 2. Laplace's equation ; steady-state temperature in a rectangular plate Page 543 3. The diffusion or heat flow equation; heat flow in a bar or slab Page 550 4. The wave equation; the vibrating string Page 554 5. Steady-state temperature in a cylinder Page 558 6. Vibration of a circular membrane Page 564 7. Steady-state temperature in a sphere Page 567 8. Poisson's equation Page 570 9. Miscellaneous problems Page 576 14. Functions of a Complex Variable Page 579 1. Introduction Page 579 2. Analytic functions Page 580 3. Contour integrals Page 588 4. Laurent series Page 592 5. The residue theorem Page 596 6. Methods of finding residues Page 598 7. Evaluation of definite integrals by use of the residue theorem Page 602 8. The point at infinity; residues at infinity Page 614 9. Mapping Page 617 10. Some applications of conformal mapping Page 622 11. Miscellaneous problems Page 630 15. Integral Transforms Page 635 1. Introduction Page 635 2. The Laplace transform Page 639 3. Solution of differential equations by Laplace transforms Page 642 4. Fourier transforms Page 647 5. Convolution; Parseval's theorem Page 655 6. Inverse Laplace transform (Bromwich integral) Page 662 7. The Dirac delta function Page 665 8. Green functions Page 670 9. Integral transform solutions of partial differential equations Page 676 10. Miscellaneous problems Page 681 16. Probability Page 685 1. Introduction; definition of probability Page 685 2. Sample space Page 687 3. Probability theorems Page 692 4. Methods of counting Page 699 5. Random variables Page 707 6. Continuous distributions Page 712 7. Binomial distribution Page 718 8. The normal or Gaussian distribution Page 723 9. The Poisson distribution Page 728 10. Applications to experimental measurements Page 731 11. Miscellaneous problems Page 737 References Page 741 Bibliography Page 743 Answers to Selected Problems Page 747 Index Page 775