Page 891 - Advanced_Engineering_Mathematics o'neil

Index 871

interior of, 374, 700–701 Cramer’s rule, 260–262
Jordan curve theorem, 700–701 defined, 247–248
line integrals and, 367–374 evaluation of, 252–258
orientation of, 374–375 matrix inverse and, 259–260
path of, 375, 380–387, 699, 701–706 matrix tree theorem, 262–264
piecewise smooth, 370–372 minor, 256
simple, 368 permutations (p), 247–248
smooth, 392–393 properties of, 249–251
Stoke’s theorem for, 408–410 Vandermark’s, 258
terminal point, 367
zero matrix elements and, 252–255
Curvilinear coordinates, 414–423
Diagonal matrices, 192, 277–283,
coordinate surfaces, 415–416
314–315
curl in, 421–423
defined, 192, 277–278
cylindrical, 414, 418–419
diagonalization, 277–283, 314–315
divergence and, 420–421
eigenvectors (E) used with, 279–283
gradient field ϕ and, 420
main, 192, 277–278
Laplacian in, 421–423
nonhomogeneous linear system
orthogonal, 416
solution using, 314–315
scale factors of, 417
off elements, 277–278
singular point of, 416
Differentiable complex function,
spherical, 415–416, 419–420
679–680
system of, 414
Differential calculus, 345–366
Cylinders, heat conduction in, 636–638
curl, 362–363, 365
Cylindrical curvilinear coordinates, 414,
curvature κ(s), 349–354
418–419
del operator ∇ for, 356–357, 362–363
divergence, 362–364
D
gradient field ϕ, 356–361
d’Alembert’s solution, wave motion and,
position vectors, 345–346
594–601
streamlines, 354–356
Damping constant (c), 61, 573–574
tangent vectors, 346–349
Data fitting, least squares vectors and,
232–236 vector analysis using, 345–366
Deformation theorem, 704, 711–712 vector fields, 354–356
Del operator ∇, 356–357, 362–363, 641 vector functions of one variable,
divergence use of, 362–363 345–349
gradient field ϕ use of, 356–657 velocity v, 349–354
Laplace’s equation use of, 641 Differential equations, 1–144, 187–246,
Dependence, see Linear dependence and 295–342, 506, 518–519, 563–666,
independence 791–793
Derivatives, Laplace transform theorem approximation solutions, 134–144
of, 82 Bernoulli, 27–28
Determinants, 247–265 Bessel functions, 114–117
cofactor expansion, 256–258 characteristic equations for, 51–54

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