Page 910 - Advanced_Engineering_Mathematics o'neil
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890 Index
Sturm-Liouville problems (Continued) Symmetry, Fourier transforms, 477
eigenfunction expansion and, Systems of differential equations,
511–515 106–110, 187–246, 295–342
eigenvalues (λ), 506–511 augmented matrix procedure for,
Fourier coefficients with respect to, 221–226
511 complex roots and, 306–308
orthogonal eigenfunctions, 511 diagonalization for, 314–315
periodic, 506–511 electrical circuits, application of,
regular, 506–507, 509 321–327
singular, 506, 509 exponential matrix solutions,
theorem for, 509 316–318
weight function p and, 511 fundamental matrix of, 300–301
Subspace S, 165–174, 177–179 general solutions of, 300, 302–306
basis of, 169–174 homogeneous systems, 213–219,
defined, 165 296–312
n
n-space R , 165–174 Laplace transform for solutions of,
orthogonal complements of, 177–179 106–110
orthogonal projection onto, 179–180 linear dependence and independence
span of, 165–166 of, 296–300, 308–312
spanning set for, 166–172 linear, 187–246, 295–342
trivial, 165 mass/spring systems, application of,
Superposition, principle of, 60 319–321
Surfaces, 388–399, 408–410
matrix operations for, 187–246,
area, integral applications of, 395 295–342
boundary curve of, 408–409 matrix solutions of, 295–302
defined, 388
nonhomogeneous systems, 220–226,
elliptical cone, 390 301–302, 312–315
fluid flux across, 397–399
phase portraits, 329–341
graphs, 388–389
reduced, 213–219
hyperbolic parabloid, 388
solution space of, 216–217
integrals, 393–399
solutions of, 295–342
level, 388
variation of parameters, method of
mass, integral applications of,
for, 312–314, 317–318
395–397
normal vector to, 389–392
piecewise smooth, 392–393 T
plane tangent to, 392 Tangent plane, 359–361, 392
simple, 389 gradient field ϕ, 359–361
smooth, 392–393 surface, to a, 392
Stoke’s theorem for, 408–410 Tangent vector, 346–349
vector integral analysis of, 388–399, Taylor method of approximation,
408–410 142–144
Symmetric matrix, 273–275 Taylor series expansion, 718–722
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