Kiselev's Calculus

Kiselev's textbooks were the standard in Russian and Soviet schools for more than half a century — and stayed there because nothing better came along to replace them. This calculus course began as Elementary Theory of Derivatives, approved by the Ministry of Education in 1908 for the final year of the Russian real schools, and was expanded a year later into Elements of Differential and Integral Calculus. It is the same Kiselev whose arithmetic, algebra, and geometry trained generations of Russian and Soviet students.

Governments changed. The math didn't. It was too good to replace.

This is the educational tradition that produced Kolmogorov, Gelfand, and Perelman — a tradition built on understanding, not memorisation. Yet Kiselev's calculus has never been properly translated into English — until now.

Why Kiselev, not another textbook?

Most modern calculus textbooks teach procedures. Kiselev teaches understanding. Every concept builds on the previous one with zero hand-waving. Every rule is derived, not dictated. Every definition is precise. Limits, derivatives, and integrals are not handed down as formulas to memorise — they are built, from the ground up, so you understand whyeach one works before you ever use it.

The structure: define → derive → apply → practise.

Modern textbooks skip the "derive" step. That's where the understanding lives.

What's inside

110 sections across six parts, building from functions to integration:

Introduction — The Concept of a Function. Explicit and implicit functions, single- and multi-valued functions, algebraic and transcendental functions, and the classification of algebraic functions. The foundation everything else rests on.

Foundations of the Theory of Limits. Infinitesimals and their properties, the precise definition of a limit, the three core theorems on limits, and limits of sums, products, quotients, powers, and roots. Limits of exponential and logarithmic functions, and the behaviour of monotone bounded variables.

Some Applications of Limits. The theory of limits put to work: rigorous proofs of the circumference and area of a circle, the volumes of the cylinder, cone, and sphere — and the construction of the number e, including the series for e and the limit of (1 + 1/n)ⁿ, leading to natural logarithms.

Beginnings of Differential Calculus. Continuity of a function and points of discontinuity. The derivative and the differential, defined rigorously and interpreted both geometrically (the tangent) and mechanically (velocity). The full toolkit of differentiation rules — sum, product, quotient, power, and root — derived as theorems. Derivatives of exponential, logarithmic, composite, inverse, power-exponential, trigonometric, and circular functions, gathered into a complete table of differentiation formulas.

Some Applications of Differential Calculus. Rolle's theorem and Lagrange's theorem. Criteria for the increase and decrease of functions. Finding greatest and least values, including of sums and products. The systematic study of how a function behaves, and the equations of the tangent and the normal.

Beginnings of Integral Calculus. The definite integral introduced through area and through distance travelled. The Fundamental Theorem — the relation between the definite integral and the antiderivative — derived, not asserted. The indefinite integral, the basic integration formulas, integration by substitution and by parts, and decomposition into summands. Applications: areas bounded by curves (parabola, ellipse, hyperbola), volumes of solids of revolution (ellipsoid and sphere), and the length of a curve.

Plus a complete set of exercises.

Who Is This Book For?

• Students taking calculus who want to actually understand it — not just survive the exam
• Self-learners rebuilding their mathematical foundations the right way
• Homeschooling families looking for a rigorous, structured calculus curriculum
• Math teachers tired of textbooks that teach tricks instead of thinking
• Anyone who already knows the formulas but was never shown where they come from
• Anyone who believes math education should be about understanding, not memorisation

About the Translator

Valery Manokhin holds a PhD in Machine Learning from Royal Holloway, University of London (supervised by Professor Vladimir Vovk), an MBA from Warwick Business School, an MSc in Computational Statistics and Machine Learning from UCL, and the Certificate in Quantitative Finance (CQF). He is the author of multiple bestselling technical books with readers in 100+ countries, the creator of the Awesome Conformal Prediction repository on GitHub, and an educator. This translation of Kiselev's Calculus is a labour of care: the goal is to make one of the finest calculus textbooks ever written accessible to English-speaking students, parents, and teachers for the first time.

About This Edition

This is the first faithful English translation of Kiselev's Elements of Differential and Integral Calculus, translated from the original Russian edition and professionally typeset in LaTeX.

This is an early-access preorder. You get the completed chapters now, and every subsequent chapter as it's finished. One purchase, full book included.

Print editions (paperback and hardcover) are available on Amazon KDP.