In this post, we will see the book Higher Mathematics For Beginners And Its Application To Physics by Ya. B. Zeldovich.

# About the book

The title of this book gives the clue to our main aim, which is to initiate the reader into the realm of differential and integral calculus and, by applying these methods to the more important divisions of physics, to demonstrate the significance and power of higher mathematics.In this book, the student is regarded as a friend and ally who puts his faith in the teacher and the textbook and wishes ardently to make use of and apply to nature and technology the mathematical techniques offered to him. Comprehension of the subject expands as the result of analyzing examples and applications. In the strictly logical approach, the question of the significance and usefulness of the theorems studied remains in the background. In the present text, by contrast, we bring to the fore the mathematical ideas and their relationship with the study of nature.The notorious pitting of poets against physicists (mathematicians too) is a figment of the imagination of the poet B. Slutsky. In mathematics there is more poetry than any poet ever imagined. The history of science is proof that good mathematics is prophetic: mathematical analysis of the known opens up the path into the realm of the unknown and leads to new physical notions.In “Higher Mathematics for Beginners” I strove towards a constructive approach, to the eliciting of the meaning and aims of mathematical concepts and attempted, at least in part, to convey the spirit of the heroic period when these notions were born.The last two chapters (Dirac’s Remarkable Delta Function and What Next) are entirely different from the remainder of the book. The style too is quite changed. The aim there is to give the reader a feeling (of necessity, very superficial) of what complicated things lie ahead.

Translated from the Russian by George Yankovsky. First published 1973, revised from the 1972 Russian edition by Mir Publishers..

PS: This is a previous version of the book by Zeldovich and Msykis we had seen earlier.

You can get the book here.

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# Contents

PREFACE TO THE FIFTH RUSSIAN EDITION 9

## CHAPTER 1 FUNCTIONS AND GRAPHS 13

The functional relationship

Coordinates

Geometric quantities expressed in terms of coordinates

Graphical representation of functions. The equation of the straight line

The parabola

The cubic parabola, hyperbola, and circle

Altering the scale of a curve

Parametric representation of a curve

## CHAPTER 2 THE CONCEPTS OF A DERIVATIVE AND AN INTEGRAL 45

Motion, distance and velocity

The derivative of a function as the limit of a ratio of increments

Notation of derivatives. The derivative of a power function

Approximating the values of a function by means of a derivative

A tangent to a curve

Increase and decrease of functions. Maximum and minimum

The area under a curve and determining distance from the rate of motion

The definite integral

The relationship between the integral and the derivative (Newton-Leibniz theorem)

The integral of a derivative

The indefinite integral

Properties of integrals

Mean values

Examples of derivatives and integrals

Summary

## CHAPTER 3 COMPUTATION OF DERIVATIVES AND INTEGRALS 106

The differential sign. The derivative of a sum of functions

The derivative of an inverse function

The composite function

The derivative of a product of functions

The power function

The derivatives of algebraic functions with constant exponents

The exponential function

The number e

Logarithms

Trigonometric functions

Inverse trigonometric functions

The derivative of an implicit function

Integrals. Statement of the problem

Elementary integrals

General properties of integrals

Change of the variable in a definite integral

Series

Computing the values of functions by means of series

Condition for applicability .of series. The geometric progression

The binomial theorem for integral and fractional exponents

The order of increase and decrease of functions

## CHAPTER 4 THE APPLICATION OF DIFFERENTIAL AND INTEGRAL CALCULUS TO GEOMETRY AND THE INVESTIGATION OF FUNCTIONS 174

Investigating maxima and minima of functions with the aid of the second derivative

Other types of maxima and minima. Salient points and discontinuities

Computing areas

Mean values

Arc length and curvature

Approximation of are length

Computing volumes, The volume and surface area of a solid of revolution

Curve sketching

## CHAPTER 5 WATER FLOW. RADIOACTIVE DECAY AND NUCLEAR FISSION. ABSORPTION OF LIGHT 211

Water flow from a vessel, Statement of the problem

The solution of an equation when the derivative depends on the desired function

Radioactive decay

Measuring the mean lifetime of radioactive atoms

Series disintegration (radioactive family)

Investigating the solution for a radioactive family (series)

The chain reaction in the fission of uranium

Multiplication of neutrons in a large system

Escape of neutrons

Critical mass

Subcritical and supercritical mass for a constant source of neutrons

The critical mass

Absorption of light. Statement of the problem and a rough estimate

The absorption equation and its solution

Relationship between exact and approximate calculations

Effective cross-section

Attenuation of a charged-particle flux of alpha and beta rays

## CHAPTER 6 MECHANICS 258

Force, work and power

Energy

Equilibrium and stability

Newton’s second law

Impulse

Kinetic energy

Motion under the action of a force dependent solely on the velocity

Motion under the action of an elastic force

Oscillations

Oscillation energy. Damped oscillations

Forced oscillations and resonance

On exact and approximate solutions of physical problems

Jet propulsion and Tsiolkovsky’s formula

The path of a projectile

The mass, centre of gravity and moment of inertia of a rod

The oscillations of a suspended rod

## CHAPTER 7 THE THERMAL MOTION OF MOLECULES AND THE DISTRIBUTION OF AIR DENSITY IN THE ATMOSPHERE 344

The condition for equilibrium in the atmosphere

The relationship between density and pressure

Density distribution

The molecular kinetic theory of density distribution

The Brownian movement and kinetic-energy distribution of molecules

Rates of chemical reactions

Evaporation. The emission current of a cathode

## CHAPTER 8 ELECTRIC CIRCUITS AND OSCILLATORY PHENOMENA IN THEM 361

Basic concepts and units of measurement

Discharge of a capacitor through a resistor

Oscillations in a capacitance circuit with spark gap

The energy of a capacitor

Inductance circuit

Breaking an inductance circuit

The energy of inductance

The oscillatory circuit

Damped oscillations

The_case of a large resistance

Alternating current

Mean quantities, power and phase shift

An alternating-current oscillatory circuit. Series resonance

Inductance and capacitance in parallel. Parallel resonance

Displacement current and the electromagnetic theory of light

Nonlinear resistance and the tunnel diode

## CHAPTER 9 DIRAC’S REMARKABLE DELTA FUNCTION 422

Various ways of defining a function

Dirac and his function

Discontinuous functions and their derivatives

Representing the delta function by formulas

Application of the delta function

CONCLUSION. WHAT NEXT? 440

ANSWERS AND SOLUTIONS 445

APPENDIX 474

INDEX 481