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Название: Mathematical Analysis
Автор: Zorich V.
Аннотация:
This softcover edition of a very popular two-volume work presents a thorough first course in analysis, leading from real numbers to such advanced topics as differential forms on manifolds, asymptotic methods, integral transforms, and distributions. Especially notable in this course is the clearly expressed orientation toward the natural sciences and its informal exploration of the essence and the roots of the basic concepts and theorems of calculus. Clarity of exposition is matched by a wealth of instructive exercises, problems and fresh applications to areas seldom touched on in real analysis books.
The second volume expounds classical analysis as it is today, as a part of unified mathematics, and its interactions with modern mathematical courses such as algebra, differential geometry, differential equations, complex and functional analysis. The book provides a firm foundation for advanced work in any of these directions.
Circulation of a field along a curve233277 Clapeyron, E.222 Class, orientation328 Clausius, R.225 Closed ball5 Closed form352 Closed set13 Closed set in a metric space56 Closed set in a topological space13 Closure613 Coefficient of thermal conductivity303 Coefficient of thermal diffusivity304 Coefficients, Fourier506—508510513519526531536541551553557 Coefficients, Fourier, extremal property506 Coefficients, Lame268275 Cohomology group356 Compact set15 Compact set in a metric space16 Compact set, elementary159 Companion trihedral73 Complete system of vectors511—516 Completeness of the trigonometric system545 Completion of a space24—27 Condition, necessary, for convergence374 Conditions Dini535570 Connected set19 Consistent charts174328 Constant, cyclic297360 Content of a set (Jordan)119 Continuity and passage to the limit383 Continuity of an improper integral depending on a parameter423—426 Continuity of an integral depending on a parameter408—409 Continuous group72336 Contribution of a maximum point616 Contribution of a point to asymptotics615 Convergence in mean528545 Convergence of a family of functions, pointwise363367 Convergence of a family of functions, uniform367 Convergence of a series of vectors505 Convergence of an improper integral153 Convergence of an improper integral, Cauchy principal value155 Convergence of distributions463 Convergence of generalized functions463 Convergence of linear functionals, strong (norm)58 Convergence of test functions463 Convergence set363367 Convergence, absolute375 Convergence, absolute, of an improper integral420 Convergence, uniform397 Convergence, uniform, Cauchy criterion369—370 Convergence, weak463 Convolution449—470 Convolution in 483 Convolution, differentiation453 Convolution, multidimensional483—493 Convolution, symmetry453 Convolution, translation-invariance453 Coordinate parallelepiped107 Coordinates of a tangent vector340 Coordinates, Cartesian259261 Coordinates, curvilinear161265 Coordinates, cylindrical268—273 Coordinates, polar166 Coordinates, spherical166268—273 Coordinates, triorthogonal268—273 Cotangent space to a manifold340 Coulomb, Ch.279288 Covering, locally finite336 Covering, refinement of another covering336 Criterion for a field to have a potential290 Criterion for compactness in a metric space17 Criterion for continuity of a mapping31 Criterion, Cauchy422 Criterion, Cauchy, for uniform convergence369—370375 Criterion, Cauchy, for uniform convergence of a series378 Criterion, Cauchy, for uniform convergence of an integral418—420 Criterion, Darboux114—116129 Criterion, Lebesgue111—114117119138145 Critical point149 Cube, singular357 Cube, singular, boundary of357 Curl203260274 Curl, physical interpretation281—282 Current function310 Curvature of a curve73 Cycle of dimension p358 Cycle, boundary358 Cycle, Carnot226 Cycles, homologous358 Cyclic constant297360 Cyclic frequency560 Cylinder167 D'Alembert, J.306 Darboux integral, lower114 Darboux integral, upper114 Darboux sum, lower114396 Darboux sum, upper114 Darboux, G.114—116120129396 de Rham, G.359 Deformation (of a closed path)293 Degree (order) of a differential form195 Delta function (-function)281449455461464469473485496559591593 Delta function (-function), shifted485 Derivation of a ring352 Derivative mapping62 Derivative of a mapping6162 Derivative of order n80 Derivative with respect to a vector81 Derivative, Lie352 Derivative, partial70—71 Derivative, second8083 Deviation, mean-square529 Diffeomorphism, elementary140 Differential equation with variables separable226 Differential form197 Differential form of class 342 Differential form of compact support344 Differential form of order zero201 Differential form on a manifold340 Differential form on a smooth surface207 Differential form, closed296352 Differential form, exact296352 Differential form, flux199 Differential form, restriction to a submanifold350 Differential form, work198 Differential of a mapping61 Differential of order n80 Differential operator486 Differential operator, adjoint486 Differential operator, self-adjoint487 Differential operator, transpose486 Differential, exterior342349 Differential, exterior, of a form201342349 Differential, partial7071 Differential, second8083 Differential, total71 Differentiation61 Differentiation at a point of a manifold347 Differentiation of a family of functions depending on a parameter388—393 Differentiation of a Fourier series544 Differentiation of a generalized function465—468 Differentiation of a power series391 Differentiation of a series391 Differentiation of an integral depending on a parameter409—412483 Differentiation of an integral, over a liquid volume497 Differentiation on a manifold347351 Differentiation with respect to a parameter426—428 Dimension of a manifold320 Dini conditions535 Dini, U.386430535570
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