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Browder A. — Mathematical Analysis: An Introduction
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Название: Mathematical Analysis: An IntroductionАвтор: Browder A. Аннотация: Among the traditional purposes of such an introductory course is the training of a student in the conventions of pure mathematics: acquiring a feeling for what is considered a proof, and supplying literate written arguments to support mathematical propositions. To this extent, more than one proof is included for a theorem - where this is considered beneficial - so as to stimulate the students' reasoning for alternate approaches and ideas. The second half of this book, and consequently the second semester, covers differentiation and integration, as well as the connection between these concepts, as displayed in the general theorem of Stokes. Also included are some beautiful applications of this theory, such as Brouwer's fixed point theorem, and the Dirichlet principle for harmonic functions. Throughout, reference is made to earlier sections, so as to reinforce the main ideas by repetition. Unique in its applications to some topics not usually covered at this level.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Год издания: 1996Количество страниц: 370Добавлена в каталог: 11.12.2011Операции: Положить на полку |
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225 235 237 270 314 (see Kronecker (see Kronecker 143 (see Sign of permutation) 237 272 211 237 204 238 205 221 (see Kronecker epsilon) A.a. (see Almost all) a.e (see Almost everywhere) Abel, N 53 72 Absolutely convergent, integral 116 Absolutely convergent, series 40 Additive, countably 206 Additive, finitely 203 Adjoint map 280 Agnesi, M 96 Alexandroff, P 153 Algebra of sets 202 Algebra, generated by 205 Almost all 221 226 Almost everywhere 226 Alternating tensor 272 Alternating tensors in 279 Alternating tensors, basis for 276 Alternating tensors, dimension of 276 Alternation operator 275 Ampere, A.M 321 Antiderivative 110 Antisymmetry 273 Arbogast, L 95 Archimedean property 15 Archimedes 26 95 Arzela, C 154 Arzela- Ascoli theorem 145 146 Ascoli, G 154 Axiom of Choice 217 218 Baire theorem 136 Baire, R 153 Ball 128 Barrow, I 122 Basis 176 Bernoulli, Jakob 53 121 Bernoulli, Johann 96 Bernstein, S 174 Bijective 3 Bolzano — Weierstrass property 144 Bolzano — Weierstrass theorem 35 Bolzano, B 53 72 153 Bonnet, O 96 Borel Cantelli lemma 222 Borel measure, translation invariant, uniqueness of 239 Borel, algebra 205 Borel, E 202 221 Borel, measure 206 Borel, set 205 214 215 220 224 Bound, lower 13 Bound, upper 13 Boundary 125 Boundary of manifold 256 Bounded 142 Bounded, totally 142 Brouwer. L.E.J 321 Bunyakovsky. V 109 Cantor function 248 Cantor set 149 201 248 Cantor set, dimension of 211 Cantor set, measure of 213 Cantor, G. 26 27 153 174 Cardinality 6 Cartan, E. 284 Cartesian product 2 133 Cauchy condensation test 43 Cauchy sequence 36 237 Cauchy. A. 72 96 121 122 Chain rule 82 184 Change of variables, linear 240 Change of variables, smooth 112 241 Class 190 Classical notation 187 Closed 124 Closed ball 257 Closure 125 Column vector 177 Compact 140 215 Compact, relatively 144 Compact, sequentially 144 Complement 2 Complement, relative 1 complete 236 Complete, metric space 130 Complete, ordered field 14 Component 148 Condensation point 150 Conjugate space 269 Connected 147 186 Connected component 148 Connected set, continuous image 148 Content 121 Continued fraction 36 39 Continuity from below 206 Continuous function 57 Continuous image of 141 Continuous map 126 Contraction map 137 193 Convergence in measure 249 Convergence, almost everywhere 249 Convergence, mean square 166 Convergence, monotone 225 Convergence, pointwise 62 226 229 Convergence, uniform 62 226 Convex coordinate patch 293 Convex function 70 78 Convex set 185 250 Convolution 251 Coordinate patch 255 Coset 218 cosets 277 Countable 6 8 Countable additivity 219 Countable base 131 Countable subadditivity 219 Countably additive 206 Countably infinite 6 Countably subadditive 209 Covector 269 283 Cramer's rule 181 Critical point 251 Critical value 251 Cross product 289 cube 242 Curl 289 Curve 253 Cut-off function 299 Daniell, P.J 252 Darboux, G 84 96 De Rham cohomology 291 Dedekind, R 21 26 Deformation 262 DeMorgan’s laws 2 212 Dense 16 126 218 220 Derivative 74 182 Derivative of convex function 80 81 Derivative of exponential 77 Derivative of polynomial 76 Derivative of trigonometric function 77 Derivative, directional 197 Derivative, higher order 88 Determinant 181 275 Determinant, Jacobian 182 Diagonal matrix 240 Diameter 142 Diffeomorphism 254 Differentiable 74 182 254 Differential 76 182 261 Differential calculus 81 Differential form 285 Differential form, closed 291 Differential form, exact 291 Differential of form (see Exterior differential) Differentiation of series 114 Dini, U 73 Dini’s theorem 64 Dirichlet principle 317 Dirichlet test for integrals 117 Dirichlet test for series 43 Dirichlet, P.G.L 53 72 174 Dirichlet’s Theorem 16 218 Disconnected 147 Disconnected, totally 149 Distribution function 248 251 Divergence 288 Divergence theorem 312 Domain 3 Dominated Convergence Theorem 230 Dual basis 270 282 Dual space 269 Dummy variable 245 Dynkin’s 238 e 33 66 Einstein, A 296 Empty set 2 Equicontinuous 145 Equivalence classes 2 217 236 Eudoxus 26 Euler, L 53 73 96 Euler’s constant 113 Expansion, binary 31 Expansion, continued fraction 38 Expansion, decimal 31 Exterior differential 287 Exterior product 277 Exterior product, (anti)commutativity of 279 Exterior product, associativity of 278 Fatou’s Lemma 230 248 Fermat, P 121 Fibonacci numbers 54 Field 10 13 Field of sets 221 Field, ordered 12 Finite induction 4 Finite intersection property 151 Finitely additive 219 Fixed point theorem, Brouwer 305 307 Fixed point theorem, contraction map 137 Fourier coefficients 164 Fourier coefficients, uniqueness of 165 Fourier series 164 Fourier, J 121 Frechet derivative 182 Frechet, M 153 182 221 Fubini Tonelli theorem 245—247 Fubini. G. 252 Function 3 16 Function, absolute value 17 Function, Borel 226 Function, Borel measurable 244 Function, characteristic 225 Function, complex 96 Function, continuous 55 57 Function, convex 70 78 96 233 250 Function, Dirichlet 56 100 Function, exponential 65—66 Function, graph of 3 251 Function, greatest integer 58 Function, harmonic 314 Function, indicator 225 Function, inverse 85 Function, Lebesgue 226 Function, limit of 55 Function, logarithm 66 Function, lower semicontinuous 151 Function, measurable 223 Function, monotone 56 Function, nowhere differentiable continuous 75 Function, periodic 162 Function, piecewise linear 61 Function, polynomial 17 58 Function, radial 320 Function, rational 17 58 Function, sign 17 Function, simple 225—228 232 Function, step 232 Function, trigonometric 66—69 Function, uniformly continuous 61 Function, upper semicontinuous 70 151 Fundamental theorem of calculus 110 111 Gauss, C.F 54 121 267 Geodesics 161 Gradient 288 Gram determinant 308 Graph 256 Grassman, H 283 Green, G 321 Green’s identities 314 Group 273 Hahn, H 221 Halmos, P 25 Hardy, G.H 27 Harmonic function 314 318 Hausdorff, dimension 211 Hausdorff, F 153 154 211 Hausdorff, measure 210 Hausdorff, space 135 Heine — Borel theorem 140 144 210 Heine, H.E 72 Hermite, C 27 Herstein, I 27 Holder, O 252 Homeomorphic 127 Homeomorphism 127 Hypersurface 265 312 IJrysohn, P 153 Image 3 178 Implicit function theorem 194 259 Implicit function theorem, classical notation 196 incommensurability 8 Indexed family 4 Induced mapping of forms 290—291 Inequality, arithmetic-geometric means 234 Inequality, Bernoulli 30 Inequality, Bessel 168 Inequality, Bunyakovsky (see Schwarz) Inequality, Cauchy 129 Inequality, Cauchy — Schwarz — Bunyakovsky 235 Inequality, Chebyshev 233 Inequality, Holder 235 Inequality, Jensen 233 Inequality, Liapounov 234 Inequality, Minkowski 235 Inequality, Schwarz 108 165
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