Kiselev's Calculus: Beginnings of Differential and Integral Calculus — First Complete English Translation (Kiselev's Mathematics (English Editions))

For most of the twentieth century, the Soviet school of mathematics produced more research-active mathematicians than any other national tradition — Kolmogorov, Arnold, Gelfand, Manin, Sinai, the Mathematical Olympiad winners, three Soviet Fields Medalists. They all learned calculus the same way. From this book.Originally published in 1909 by A. P. Kiselev as the calculus companion to his Elementary Algebra and Elementary Geometry — the textbooks from which Russian and Soviet schools taught mathematics for over 70 years — Elements of Differential and Integral Calculus became the standard first encounter with the subject across the Russian and Soviet educational system. Now, for the first time, the complete English translation is available.What makes Kiselev's Calculus different from every modern textbook:The limit is built before the derivative. Modern American calculus textbooks define the derivative first, then introduce limits as the machinery underneath. Kiselev does the opposite. Chapter 2 develops the theory of limits as a self-contained subject — infinitesimals, the algebra of limits, the limit of a monotone bounded sequence, the sandwich theorem — before any derivative is mentioned. The student arrives at the derivative with a working concept of limit already in hand. This is the historical order of the subject, and it remains the cleanest order pedagogically.Every theorem is proved. Where modern textbooks state results and refer the curious reader to an appendix, Kiselev proves them in the body of the text: the continuity of the elementary functions, the derivative of xⁿ for rational n, the derivative of an inverse function, Rolle's theorem, Lagrange's Mean Value Theorem, the criterion for monotonicity, the criterion for maxima and minima using higher derivatives, the existence of the definite integral as the limit of Riemann sums, and the Fundamental Theorem of Calculus. The proofs are accessible to a student finishing secondary algebra — they need no abstract machinery beyond what Kiselev builds in earlier chapters.The proof of lim(sin α)/α = 1 — in chapter 3. The most-cited "trust me" limit in calculus is proved here in five lines, from a circular arc with an inscribed chord and a circumscribed tangent. A 17-year-old can follow it.The integral is developed in parallel with two physical problems at once. Chapter 6 introduces the definite integral not as an antiderivative computation, but as the common limit of two complementary constructions — the area under a curve and the distance covered by a moving body. The integral concept is concrete before any formal calculation. Then Chapter 7 develops the rules: substitution, integration by parts, partial fractions, integrals of rational and irrational functions.147 pages. 7 chapters. 110 sections. 41 figures. 214 exercises with answers.This translation preserves Kiselev's original section numbering, the order of his exposition, and the precision of his prose. The typography is modern; the mathematics is Kiselev's. Notation is presented as Kiselev presented it: Log for the natural logarithm, tg and cotg for the tangent and cotangent, each with explanatory footnotes on first use.Ideal for: homeschool families using a rigorous mathematics curriculum, students preparing for university calculus who want a proof-based first encounter with the subject, teachers of advanced-placement or olympiad calculus, undergraduates rebuilding their first calculus before tackling real analysis, adults retracing the mathematical education they were never given, and anyone interested in the tradition that produced the Russian school of mathematics.Translated by Valery Manokhin, PhD, MBA, CQF — mathematician, machine-learning researcher, and translator of English editions of classic Russian mathematics textbooks. Read more