Mathematical Methods for Physics and Engineering - Matematica.NET

INDEXgradient operator (grad), 348–352, 367identities, 356, 978Laplacian, 352, 368non-Cartesian, 357–369tensor forms, 971–975curl, 974divergence, 972gradient, 972Laplacian, 973vector product, 222–224anticommutativity, 222definition, 222determinant form, 224in Cartesian coordinates, 224non-associativity, 222vector spaces, 242–247, 1113associativity of addition, 242basis vectors, 243commutativity of addition, 242complex, 242defining properties, 242dimensionality, 243group actions on, 1088inequalities: Bessel, Schwarz, triangle, 246invariant, 1088, 1113matrices as an example, 252of infinite dimensionality, 556–559associativity of addition, 556basis functions, 556commutativity of addition, 556defining properties, 556Hilbert spaces, 557–559inequalities: Bessel, Schwarz, triangle, 559parallelogram equality, 247real, 242span of a set of vectors in, 242vector triple product, 226identities, 226non-associativity, 226vectorsas first-order tensors, 932as geometrical objects, 241base, 336column, 250compared with scalars, 212component form, 217examples of, 212graphical representation of, 212irrotational, 353magnitude of, 218non-Cartesian, 336, 358, 362notation, 212polar and axial, 949solenoidal, 352, 389span of, 242vectors, algebra of, 212–234addition and subtraction, 213in component form, 218angle between, 221associativity of addition and subtraction, 213commutativity of addition and subtraction,213multiplication by a complex scalar, 222multiplication by a scalar, 214multiplication of, see scalar product andvector productouter product, 936vectors, applicationscentroid of a triangle, 216equation of a line, 226equation of a plane, 227equation of a sphere, 228finding distance from aline to a line, 231line to a plane, 232point to a line, 229point to a plane, 230intersection of two planes, 228vectors, calculus of, 334–369differentiation, 334–339, 344integration, 339line integrals, 377–389surface integrals, 389–396volume integrals, 396vectors, derived quantitiescurl, 353derivative, 334differential, 338, 344divergence (div), 352reciprocal, 233, 366, 955, 959vector fields, 347curl, 406divergence, 352flux, 395rate of change, 350vectors, physicalacceleration, 335angular momentum, 238angular velocity, 223, 238, 353area, 393–395, 408area of parallelogram, 223, 224force, 212, 213, 220moment or torque of a force, 223velocity, 335velocity vectors, 335Venn diagrams, 1119–1124vibrationsinternal, see normal modeslongitudinal, in a rod, 677transversemembrane, 677, 739, 768, 799, 801rod, 769string, 676, 789Volterra integral equation, 804, 805differentiation methods, 812Laplace transform methods, 810volume elementscurvilinear coordinates, 365cylindrical polars, 360spherical polars, 205, 3621332