INDEX 677

lim inf, 113lim sup, 113liminf, 86limit comparison test, 157limit of a function, 317limit of a subsequence, 85limit of nth root of n, 243limit point, 91, 447limit points

closed sets, 92limitfore, 88limits

at infinity, 114properties, 116squeezing theorem, 84, 117uniqueness, 82, 115well defined, 115

limits and continuity, 319limits of sequences

preservation of order, 85properties, 83

limsup, 86line integral, 516linear combination, 374, 398, 436linear functions, 373linear initial value problem, 185linear map, 373linear transformation, 373lines

parametric equation, 276Lipschitz, 113, 326, 327lizards

surface area, 558local extrema, 132local extremum, 487

derivative equals 0, 132local maximum, 132, 487local minimum, 132, 487locating loca extreemum, 487logistic function, 76Lotka Volterra equations, 321lower semicontinuous, 113lower sums, 68, 178

main diagonal, 412mass balance, 579material coordinates, 580mathematical induction, 26

matriceseigenvalues exist, 418

matrixleft inverse, 441lower triangular, 412right inverse, 441upper triangular, 412

maximizing sequence, 113mean value theorem

Cauchy, 138usual version, 138

measurabilitylimit of simple functions, 636

measurable, 638measurable sets, 639measurable space, 633measure, 633

properties, 633measure space, 633measures

decreasing sequences of sets, 633increasing sequences of sets, 633

measures from outer measures, 639Merten’s theorem, 166, 256metric tensor, 559, 614, 616metric tensor, 625minimal polynomial, 407minimizing sequence, 113minor, 410, 411, 438mixed partial derivatives, 454moment of a force, 298monic

polynomial, 36monotone convergence theorem, 646motion, 580multi-index, 316multiplicative inverse

unique, 16

Navier, 593nested interval lemma, 87Neuman series, 397Newton Ralphson procedure, 146Newton’s second law, 620nonremovable discontinuity, 97normal vector to plane, 303

one to one