405
INDEX Partial differential equations (cont.) elliptic, 258, 261 existence and uniqueness theorems for, 258 general solution of, 258, 262, 263, 277 homogeneous, 258 hyperbolic, 258, 261 linear, 258, 261 non-homogeneous, 258 non-linear, 258, 261 order of, 258 parabolic, 258, 261 particular solution of, 258, 262 singular solutions of, 258 solutions of, 258, 262-264 sone important, 259, 260, 263, 264 Partial fractions, 84 used in finding inverse Laplace transforms, 102,113,114 Partial sums, 6 Particular solution, 39, 45, 46, 258, 262, 263 of a linear differential equation, 72-74, 260 Pendulum, 378 Period, 57,182 Periodic functions, 2, 182 Laplace transforms of, 101 Perpendicular vectors, condition for, 124 Picard's method, 44, 63, 64 Piecewise continuity, 99,105 Plane, equation of, 135 Planets, motion of, 389 Points, on a line, 1 Polar coordinates, 158, 168 Polar form of complex numbers, 12, 29, 30 Polar moment of inertia, 155 Poles, 288, 289 Polynomials, 2 Position vector, 123 Positive integers, 1 Positive normal, 154 Positive numbers, 1 Positive sense for traversing a curve, 151,162 Postulates, 2 Potential, 42, 377, 378, 387 equation, 259 function, 377, 378 of a hollow sphere, 276, 277 Potential energy, principle of minimum, 394 Power series, 8 uniform convergence of, 8 Primitive, 39 Principal branch, 295 Principal normal vector to a space curve, 137, 144 Principal part of a Laurent series, 289 Principal value of logarithms, 295 Product, cross [see Cross product] dot [see Dot product] of matrices, 343 Quadratic equation, 29 Quadratic form, 349, 367, 368, 370 in n variables, 370
Quadratic form (cont.) reduction of to canonical form, 349, 368 symmetric, 370 Quotient, 1 Radioactivity, 43 Radium, decay of, 60 Radius of curvature, 137,144 Radius of torsion, 144 Radius vector, 123 Ratio test, 7 Rational numbers, 1 Rayleigh-Ritz methods, 379, 390-392 Rayleigh's principle, 391 Real numbers, 1,13 Real part, of a complex number, 12, 286 Rectangular component vectors, 123 Rectangular coordinate system, right handed, 122 Rectangular coordinates, 128 Rectified sine wave, 111 Recurrence or recursion formulas, 77 for Bessel functions, 225, 230 for gamma function, 210 for Hermite polynomials, 244 for Laguerre polynomials, 245 for Legendre polynomials, 248 for modified Bessel functions, 235 Reduced equation, 71, 80, 81 Reduction of order, method of, 75, 77, 81 Region of convergence of series, 7 Relative maxima and minima, 11 [see also Maxima and minima] Remainder, in Taylor series, 8 Removable singularity, 289, 303 Residue theorem, 289, 290, 304-306 evaluation of integrals by, 306-311 proof of, 304, 305 use of in finding inverse Laplace transforms, 324, 328-338 Residues, 289, 304-306 [see also Residue Theorem] Resistance, 42 Resistor, 42 Resonance, 91 Resultant of vectors, 121 Riemann-Cauchy equations [see Cauchy-Riemann equations] Riemann's mapping theorem, 291, 292 Riemann's theorem, 195, 207 Right and left hand limits, 183 Right handed coordinate system, 123 Rodrigue's formula, for Hermite polynomials, 244 for Laguerre polynomials, 244 for Lagendre polynomials, 242, 247 Roots, of complex numbers, 12 of polynomial equations, 2 Rotation, 292 expressed in matrix form, 350 Rules of algebra, 1 Runge-Kutta method, 44, 64, 65 Saturated solutions, 60 Scalar field, 125