Elementary Illustrations of the Differential and Integral Calculus (Illustrated) Kindle Edition

DIFFERENTIAL AND INTEGRAL CALCULUS.ELEMENTARY ILLUSTRATIONS.The Differential and Integral Calculus, or, as it was formerly called, the Doctrine of Fluxions, has always been supposed to present remarkable obstacles to the beginner. It is matter of common observation that anyone who commences this study, even with the best elementary works, finds himself in the dark as to the real meaning of the processes which he learns, until, at a certain stage of his progress, depending upon his capacity, some accidental combination of his own ideas throws light upon the subject. The reason of this may be that it is usual to introduce him at the same time to new principles, processes, and symbols, thus preventing his attention from being exclusively directed to one new thing at a time. It is our belief that this should be avoided; and we propose, therefore, to try the experiment, whether by undertaking the solution of some problems by common algebraic methods, without calling for the reception of more than one new symbol at once, or lessening the immediate evidence of each investigation by reference to general rules, the study of more methodical treatises may not be somewhat facilitated. We would not, nevertheless, that the student should imagine we can remove all obstacles; we must introduce notions, the consideration of which has not hitherto occupied his mind; and shall therefore consider our object as gained, if we can succeed in so placing the subject before him, that two independent difficulties shall never occupy his mind at once.CONTENTS:On the Ratio or Proportion of Two MagnitudesOn the Ratio of Magnitudes that Vanish TogetherOn the Ratios of Continuously Increasing or Decreasing QuantitiesThe Notion of Infinitely Small QuantitiesOn FunctionsInfinite SeriesConvergent and Divergent SeriesTaylor's Theorem Derived FunctionsDifferential CoefficientsThe Notation of the Differential CalculusAlgebraic GeometryOn the Connexion of the Signs of Algebraic and the Directions of Geometrical MagnitudesThe Drawing of a Tangent to a CurveRational Explanation of the Language of LeibnitzOrders of InfinityA Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight LinesThe Same Problem Solved by the Principles of LeibnitzAn Illustration from Dynamics: Velocity, Acceleration, etc.Simple Harmonic MotionThe Method of FluxionsAccelerated Motion Limiting Ratios of Magnitudes that Increase Without LimitRecapitulation of Results Reached in the Theory of FunctionsApproximations by the Differential CalculusSolution of Equations by the Differential CalculusPartial and Total DifferentialsApplication of the Theorem for Total Differentials to the Determination of Total Resultant ErrorsRules for DifferentiationIllustration of the Rules for DifferentiationDifferential Coefficients of Differential CoefficientsCalculus of Finite Differences Successive DifferentiationTotal and Partial Differential Coefficients Implicit DifferentiationApplications of the Theorem for Implicit DifferentiationInverse FunctionsImplicit FunctionsFluxions and the Idea of TimeThe Differential Coefficient Considered with Respect to its MagnitudeThe Integral CalculusConnexion of the Integral with the Differential CalculusNature of IntegrationDetermination of Curvilinear Areas the ParabolaMethod of IndivisiblesConcluding Remarks on the Study of the CalculusBibliography of Standard Text-books and Works of Reference on the Calculus Read more