Page 905 - Advanced_Engineering_Mathematics o'neil

Index 885

P improper node, 334–336
Parallelogram law, vector applications nodal sink, 331
of, 149–150, 152 nodal source, 331–332
Parametric equations, 3-space vector prey/predator model, application of,
lines, 152–154 338–340
Parseval’s inequality, 175 proper node, 333–334
Parseval’s theorem, 450–451, 515–518 saddle point, 333
eigenfunction expansions, 515–518 spiral point, 335
Fourier series integration using, spiral sink, 335
450–451 spiral source, 335–336
Partial differential equations, 563–666 Piecewise continuity, 81–82
heat equation, 611–639 Piecewise continuous functions,
Laplace’s equation, 641–644 431–432
potential equation, 641–666 Piecewise smooth functions, 432
wave equation, 565–610 Piecewise smooth surface, 392–393
Partial fractions decomposition, Laplace Plane-parallel flow, 779–780
transform and, 84, 118–120 Planes, 380–387, 392, 670, 779–786
Particular solutions, 4–6 axes, real and imaginary, 670
Path of a curve, 375, 380–387, 699, complex, 670
701–706 conformal mapping of, 779–786
Cauchy’s theorem and, 701–706 conservative vector fields in, 380–387
closed, 375, 381–382 domain D, 383–387
complex function integrals and, 699, fluid flow models, 779–786
701–706 independence of path and, 383–387
defined, 375, 701 potential theory and, 383–387
deformation theorem and, 704–706 tangent to a surface, 392
independence of, 380–387, 699, vector integral analysis of, 380–387,
703–704 392
potential theory and, 380–387 Plates, heat conduction in, 638–639
Periodic boundary conditions, 506 Poisson’s integral formula, 561,
Periodicity, 494, 605–606 648–649
discrete Fourier transform (DFT), 494 Polar form, 457, 672–673, 730–738
vibration, 605–606 argument and, 672–673
Permutations (p), 247–248 Fourier series, 457
Phase angle form, Fourier series, pole of order m, 730–732, 736–738
452–456 poles of quotients, 732–733
Phase portraits, 329–341 residues at, 734–738
center of system, 337 simple, 734–736
competing species model, application singularities, 730–733
of, 340–341 Polynomial coefficients, 112–117,
defined, 330 269–273
eigenvalue (λ) classification of, Bessel functions, 114–117
329–338 characteristic, 269–271

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