INDEX 693

lim inf, 118lim sup, 118liminf, 88limit comparison test, 165limit of a function, 325limit of a subsequence, 87limit of nth root of n, 251limit point, 93, 459limit points

closed sets, 94limitfore, 90limits

at infinity, 119properties, 121squeezing theorem, 86, 121uniqueness, 84, 120well defined, 120

limits and continuity, 327limits of sequences

preservation of order, 87properties, 85

limsup, 88line integral, 528linear combination, 386, 410, 448linear functions, 385linear initial value problem, 193linear map, 385linear transformation, 385lines

parametric equation, 284Lipschitz, 117, 334, 335lizards

surface area, 572local extrema, 139local extremum, 499

derivative equals 0, 139local maximum, 139, 499local minimum, 139, 499locating loca extreemum, 499logistic function, 77Lotka Volterra equations, 329lower semicontinuous, 118lower sums, 69, 186

main diagonal, 424mass balance, 594material coordinates, 594mathematical induction, 26

matriceseigenvalues exist, 430

matrixleft inverse, 453lower triangular, 424right inverse, 453upper triangular, 424

maximizing sequence, 118mean value theorem

Cauchy, 144usual version, 145

measurabilitylimit of simple functions, 650

measurable, 652measurable sets, 653measurable space, 647measure, 647

properties, 647measure space, 647measures

decreasing sequences of sets, 647increasing sequences of sets, 647

measures from outer measures, 653Merten’s theorem, 174, 264metric tensor, 573, 628, 630metric tensor, 639minimal polynomial, 419minimizing sequence, 118minor, 422, 423, 450mixed partial derivatives, 466moment of a force, 306monic

polynomial, 36monotone convergence theorem, 660motion, 595multi-index, 324multiplicative inverse

unique, 16

Navier, 607nested interval lemma, 89Neuman series, 409Newton Ralphson procedure, 152Newton’s second law, 634nonremovable discontinuity, 101normal vector to plane, 311

one to one