Multivariable Advanced Calculus

INDEX 463measurable, 165measurable function, 175pointwise limits, 175measurable functionsBorel, 201measurable sets, 165measure, 157measure space, 157minimal polynomial, 50minor, 43mixed partial derivatives, 135mollifier, 220monotone convergence theorem, 184monotone functionsdifferentiable, 252multi - index, 98multi-index, 132nested interval lemma, 74nonlinear Fubini’s theorem, 457nonmeasurable set, 205normp norm, 55odd map, 269open cover, 153open set, 75orientable manifold, 295orientation, 363oriented curve, 363orthonormal, 58outer measure, 157, 201outer regularity, 159parallelepiped, 393partial derivatives, 124pi systems, 207pointwise convergencesequence, 95series, 97Poisson’s equation, 327Poisson’s integral formula, 335Poisson’s problem, 327polar decompositionleft, 68right, 65potential, 374power set, 11precompact, 118primitive, 401probability measure, 179probability space, 179product formula, 277rank of a matrix, 46rational function, 98real part, 397rectifiable, 361rectifiable curve, 361regular measure, 159regular values, 259relative topology, 293retraction, 247right Cauchy Green strain tensor, 65right handed system, 392Rouche theorem, 414Russell’s paradox, 13Sard’s lemma, 241scalars, 25Schroder Bernstein theorem, 14Schur’s theorem, 62second derivative test, 146self adjoint, 60separable, 83separated, 90sets, 11sigma algebra, 157simple curve, 361simple functions, 178singular values, 259smooth surface, 395span, 26Steiner symetrization, 427Stirling’s formula, 203Stoke’s theorem, 391, 396Stokes theorem, 311subharmonic, 341subspace, 26support, 195Taylor’s formula, 144Tietze extension theorem, 106triangle inequality, 55trivial, 26uniform contractions, 138uniform convergencesequence, 95series, 97uniformly Cauchysequence, 95uniformly continuous, 94uniformly integrable, 202, 298unitary, 66upper semicontinuous, 116vector space axioms, 25