Synopsis
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 Magnitudes On the Ratio of Magnitudes that Vanish Together On the Ratios of Continuously Increasing or Decreasing Quantities The Notion of Infinitely Small Quantities On Functions Infinite Series Convergent and Divergent Series Taylor's Theorem Derived Functions Differential Coefficients The Notation of the Differential Calculus Algebraic Geometry On the Connexion of the Signs of Algebraic and the Directions of Geometrical Magnitudes The Drawing of a Tangent to a Curve Rational Explanation of the Language of Leibnitz Orders of Infinity A Geometrical Illustration: Limit of the Intersections of Two Coinciding Straight Lines The Same Problem Solved by the Principles of Leibnitz An Illustration from Dynamics: Velocity, Acceleration, etc. Simple Harmonic Motion The Method of Fluxions Accelerated Motion Limiting Ratios of Magnitudes that Increase Without Limit Recapitulation of Results Reached in the Theory of Functions Approximations by the Differential Calculus Solution of Equations by the Differential Calculus Partial and Total Differentials Application of the Theorem for Total Differentials to the Determination of Total Resultant Errors Rules for Differentiation Illustration of the Rules for Differentiation Differential Coefficients of Differential Coefficients Calculus of Finite Differences Successive Differentiation Total and Partial Differential Coefficients Implicit Differentiation Applications of the Theorem for Implicit Differentiation Inverse Functions Implicit Functions Fluxions and the Idea of Time The Differential Coefficient Considered with Respect to its Magnitude The Integral Calculus Connexion of the Integral with the Differential Calculus Nature of Integration Determination of Curvilinear Areas the Parabola Method of Indivisibles Concluding Remarks on the Study of the Calculus Bibliography of Standard Text-books and Works of Reference on the Calculus
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