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The Advanced calculus problem solver

The Advanced calculus problem solver - Max Fogiel, Research And Education Association
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Each Problem Solver is an insightful and essential study and solution guide chock-full of clear, concise problem-solving gems. All your questions can be found in one convenient source from one of the most trusted names in reference solution guides. More useful, more practical, and more informative, these study aids are the best review books and textbook companions available. Nothing remotely as comprehensive or as helpful exists in their subject anywhere. Perfect for undergraduate and graduate studies.Here in this highly useful reference is the finest overview of advanced calculus currently available, with hundreds of calculus problems that cover everything from point set theory and vector spaces to theories of differentiation and integrals. Each problem is clearly solved with step-by-step detailed solutions.DETAILS- The PROBLEM SOLVERS are unique - the ultimate in study guides. - They are ideal for helping students cope with the toughest subjects. - They greatly simplify study and learning tasks. - They enable students to come to grips with difficult problems by showing them the way, step-by-step, toward solving problems. As a result, they save hours of frustration and time spent on groping for answers and understanding. - They cover material ranging from the elementary to the advanced in each subject. - They work exceptionally well with any text in its field. - PROBLEM SOLVERS are available in 41 subjects. - Each PROBLEM SOLVER is prepared by supremely knowledgeable experts. - Most are over 1000 pages. - PROBLEM SOLVERS are not meant to be read cover to cover. They offer whatever may be needed at a given time. An excellent index helps to locate specific problems rapidly. - Educators consider the PROBLEM SOLVERS the most effective and valuable study aids; students describe them as "fantastic" - the best books on the market.TABLE OF CONTENTSIntroductionChapter 1: Point Set TheorySets and SequencesClosed and Open Sets and NormsMetric SpacesChapter 2: Vector SpacesDefinitionsPropertiesInvertibilityDiagonalizationOrthogonalityChapter 3: ContinuityShowing that a Function is ContinuousDiscontinuous FunctionsUniform Continuity and Related TopicsParadoxes of ContinuityChapter 4: Elements of Partial DifferentiationPartial DerivativesDifferentials and the JacobianThe Chain RuleGradients and Tangent PlanesDirectional DerivativesPotential FunctionsChapter 5: Theorems of DifferentiationMean Value TheoremsTaylor's TheoremImplicit Function TheoremChapter 6: Maxima and MinimaRelative Maximum and Relative MinimumExtremes Subject to a ConstraintExtremes in a RegionMethod of Lagrange MultipliersFunctions of Three VariablesExtreme Value in RnChapter 7: Theory of IntegrationRiemann IntegralsStieltjes IntegralsChapter 8: Line IntegralsMethod of ParametrizationMethod of Finding Potential Function (Exact Differential)Independence of PathGreen's TheoremChapter 9: Surface Integrals Change of Variables FormulaAreaIntegral Function over a SurfaceIntegral Vector Field over a SurfaceInvergence TheoremStoke's TheoremDifferential FormChapter 10: Improper IntegralsImproper Integrals of the 1st, 2nd, and 3rd KindAbsolute and Uniform ConvergenceEvaluation of Improper IntegralsGamma and Beta FunctionsChapter 11: Infinite SequencesConvergence of SequencesLimit Superior and Limit InferiorSequence of FunctionsChapter 12: Infinite SeriesTests for Convergence and DivergenceSeries of FunctionsOperations on SeriesDifferentiation and Integration of SeriesEstimates of Error and SumsCesaro SummabilityInfinite ProductsChapter 13: Power SeriesInterval of ConvergenceOperations on Power SeriesChapter 14: Fourier SeriesDefinitions and ExamplesConvergence QuestionsFurther RepresentationsApplicationsChapter 15: Complex VariablesComplex NumbersComplex Functions and DifferentiationSeriesIntegrationChapter 16: Laplace TransformsDefinitions and Simple ExamplesBasic Properties of Laplace TransformsStep Functions and Periodic FunctionsThe Inversion ProblemApplicationsChapter 17: Fourier TransformsDefinition of Fourier TransformsProperties of Fourier TransformsApplications of Fourier TransformsChapter 18: Differential GeometryCurvesSurfacesChapter 19: Miscellaneous Problems and ApplicationsMiscellaneous ApplicationsElliptic IntegralsPhysical ApplicationsIndexWHAT THIS BOOK IS FORStudents have generally found calculus a difficult subject to understand and learn. Despite the publication of hundreds of textbooks in this field, each one intended to provide an improvement over previous textbooks, students of advanced calculus continue to remain perplexed as a result of numerous subject areas that must be remembered and correlated when solving problems. Various interpretations of advanced calculus terms also contribute to the difficulties of mastering the subject.In a study of calculus, REA found the following basic reasons underlying the inherent difficulties of advanced calculus:No systematic rules of analysis were ever developed to follow in a step-by-step manner to solve typically encountered problems. This results from numerous different conditions and principles involved in a problem that leads to many possible different solution methods. To prescribe a set of rules for each of the possible variations would involve an enormous number of additional steps, making this task more burdensome than solving the problem directly due to the expectation of much trial and error.Current textbooks normally explain a given principle in a few pages written by a mathematician who has insight into the subject matter not shared by others. These explanations are often written in an abstract manner that causes confusion as to the principle's use and application. Explanations then are often not sufficiently detailed or extensive enough to make the reader aware of the wide range of applications and different aspects of the principle being studied. The numerous possible variations of principles and their applications are usually not discussed, and it is left to the reader to discover this while doing exercises. Accordingly, the average student is expected to rediscover that which has long been established and practiced, but not always published or adequately explained.The examples typically following the explanation of a topic are too few in number and too simple to enable the student to obtain a thorough grasp of the involved principles. The explanations do not provide sufficient basis to solve problems that may be assigned for homework or given on examinations.Poorly solved examples such as these can be presented in abbreviated form which leaves out much explanatory material between steps, and as a result requires the reader to figure out the missing information. This leaves the reader with an impression that the problems and even the subject are hard to learn - completely the opposite of what an example is supposed to do.Poor examples are often worded in a confusing or obscure way. They might not state the nature of the problem or they present a solution, which appears to have no direct relation to the problem. These problems usually offer an overly general discussion - never revealing how or what is to be solved.Many examples do not include accompanying diagrams or graphs, denying the reader the exposure necessary for dr


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