INDEX
gradient operator (grad), 348–352, 367
identities, 356, 978
Laplacian, 352, 368
non-Cartesian, 357–369
tensor forms, 971–975
curl, 974
divergence, 972
gradient, 972
Laplacian, 973
vector product, 222–224
anticommutativity, 222
definition, 222
determinant form, 224
in Cartesian coordinates, 224
non-associativity, 222
vector spaces, 242–247, 1113
associativity of addition, 242
basis vectors, 243
commutativity of addition, 242
complex, 242
defining properties, 242
dimensionality, 243
group actions on, 1088
inequalities: Bessel, Schwarz, triangle, 246
invariant, 1088, 1113
matrices as an example, 252
of infinite dimensionality, 556–559
associativity of addition, 556
basis functions, 556
commutativity of addition, 556
defining properties, 556
Hilbert spaces, 557–559
inequalities: Bessel, Schwarz, triangle, 559
parallelogram equality, 247
real, 242
span of a set of vectors in, 242
vector triple product, 226
identities, 226
non-associativity, 226
vectors
as first-order tensors, 932
as geometrical objects, 241
base, 336
column, 250
compared with scalars, 212
component form, 217
examples of, 212
graphical representation of, 212
irrotational, 353
magnitude of, 218
non-Cartesian, 336, 358, 362
notation, 212
polar and axial, 949
solenoidal, 352, 389
span of, 242
vectors, algebra of, 212–234
addition and subtraction, 213
in component form, 218
angle between, 221
associativity of addition and subtraction, 213
commutativity of addition and subtraction,
213
multiplication by a complex scalar, 222
multiplication by a scalar, 214
multiplication of,seescalar productand
vector product
outer product, 936
vectors, applications
centroid of a triangle, 216
equation of a line, 226
equation of a plane, 227
equation of a sphere, 228
finding distance from a
line to a line, 231
line to a plane, 232
point to a line, 229
point to a plane, 230
intersection of two planes, 228
vectors, calculus of, 334–369
differentiation, 334–339, 344
integration, 339
line integrals, 377–389
surface integrals, 389–396
volume integrals, 396
vectors, derived quantities
curl, 353
derivative, 334
differential, 338, 344
divergence (div), 352
reciprocal, 233, 366, 955, 959
vector fields, 347
curl, 406
divergence, 352
flux, 395
rate of change, 350
vectors, physical
acceleration, 335
angular momentum, 238
angular velocity, 223, 238, 353
area, 393–395, 408
area of parallelogram, 223, 224
force, 212, 213, 220
moment or torque of a force, 223
velocity, 335
velocity vectors, 335
Venn diagrams, 1119–1124
vibrations
internal,seenormal modes
longitudinal, in a rod, 677
transverse
membrane, 677, 739, 768, 799, 801
rod, 769
string, 676, 789
Volterra integral equation, 804, 805
differentiation methods, 812
Laplace transform methods, 810
volume elements
curvilinear coordinates, 365
cylindrical polars, 360
spherical polars, 205, 362