## Higher Mathematics For Beginners And Its Application To Physics – Zeldovich

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

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
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