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El. knyga: Mathematics for Natural Scientists: Fundamentals and Basics

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This book, now in a second revised and enlarged edition, covers a course of mathematics designed primarily for physics and engineering students. It includes all the essential material on mathematical methods, presented in a form accessible to physics students and avoiding unnecessary mathematical jargon and proofs that are comprehensible only to mathematicians. Instead, all proofs are given in a form that is clear and sufficiently convincing for a physicist. Examples, where appropriate, are given from physics contexts. Both solved and unsolved problems are provided in each section of the book. The second edition includes more on advanced algebra, polynomials and algebraic equations in significantly extended first two chapters on elementary mathematics, numerical and functional series and ordinary differential equations. Improvements have been made in all other chapters, with inclusion of additional material, to make the presentation clearer, more rigorous and coherent, and the number of problems has been increased at least twofold. Mathematics for Natural Scientists: Fundamentals and Basics is the first of two volumes. Advanced topics and their applications in physics are covered in the second volume the second edition of which the author is currently being working on.

Part I Fundamentals
1 Basic Knowledge
3(180)
1.1 Logic of Mathematics
3(2)
1.2 Real Numbers
5(6)
1.2.1 Integers
6(1)
1.2.2 Rational Numbers
7(2)
1.2.3 Irrational Numbers
9(1)
1.2.4 Real Numbers
10(1)
1.2.5 Intervals
11(1)
1.3 Basic Tools: An Introduction
11(33)
1.3.1 Cartesian Coordinates in 2D and 3D Spaces
11(1)
1.3.2 Algebra
12(7)
1.3.3 Inequalities
19(2)
1.3.4 Functions
21(3)
1.3.5 Simple Algebraic Equations
24(8)
1.3.6 Systems of Algebraic Equations
32(8)
1.3.7 Functional Algebraic Inequalities
40(4)
1.4 Polynomials
44(18)
1.4.1 Division of Polynomials
44(5)
1.4.2 Finding Roots of Polynomials with Integer Coefficients
49(4)
1.4.3 Vieta's Formulae
53(4)
1.4.4 Factorisation of Polynomials: Method of Undetermined Coefficients
57(3)
1.4.5 Multiplication of Polynomials
60(2)
1.5 Elementary Geometry
62(4)
1.5.1 Circle, Angles, Lines, Intersections, Polygons
62(3)
1.5.2 Areas of Simple Plane Figures
65(1)
1.6 Trigonometric Functions
66(7)
1.7 Golden Ratio and Golden Triangle. Fibonacci Numbers
73(4)
1.8 Essential Smooth 2D Curves
77(5)
1.9 Simple Determinants
82(3)
1.10 Vectors
85(20)
1.10.1 Three-Dimensional Space
85(12)
1.10.2 N-Dimensional Space
97(1)
1.10.3 My Father's Number Pyramid
98(7)
1.11 Introduction to Complex Numbers
105(13)
1.11.1 Cardano's Formula
105(3)
1.11.2 Complex Numbers
108(4)
1.11.3 Square Root of a Complex Number
112(4)
1.11.4 Polynomials with Complex Coefficients
116(1)
1.11.5 Factorisation of a Polynomial with Real Coefficients
117(1)
1.12 Summation of Finite Series
118(4)
1.13 Binomial Formula
122(6)
1.14 Summae Potestatum and Bernoulli Numbers
128(3)
1.15 Prime Numbers
131(3)
1.16 Combinatorics and Multinomial Theorem
134(5)
1.17 Elements of Classical Probability Theory
139(11)
1.17.1 Trials, Outcomes and Sets
140(2)
1.17.2 Definition of Probability of a Random Event
142(2)
1.17.3 Main Theorems of Probability
144(6)
1.18 Some Important Inequalities
150(10)
1.18.1 Cauchy-Bunyakovsky-Schwarz Inequality
150(3)
1.18.2 Angles Inequality
153(1)
1.18.3 Four Averages of Positive Numbers
154(6)
1.19 Lines, Planes and Spheres
160(23)
1.19.1 Straight Lines
161(1)
1.19.2 Polar and Spherical Coordinates
162(2)
1.19.3 Curved Lines
164(2)
1.19.4 Planes
166(1)
1.19.5 Circle and Sphere
167(1)
1.19.6 Typical Problems for Lines, Planes and Spheres
168(15)
2 Functions
183(60)
2.1 Definition and Main Types of Functions
183(5)
2.2 Infinite Numerical Sequences
188(7)
2.2.1 Definitions
188(2)
2.2.2 Main Theorems
190(4)
2.2.3 Sum of an Infinite Numerical Series
194(1)
2.3 Elementary Functions
195(25)
2.3.1 Polynomials
196(1)
2.3.2 Rational Functions
196(8)
2.3.3 General Power Function
204(3)
2.3.4 Number e
207(3)
2.3.5 Exponential Function
210(1)
2.3.6 Hyperbolic Functions
211(1)
2.3.7 Logarithmic Function
212(1)
2.3.8 Trigonometric Functions
213(5)
2.3.9 Inverse Trigonometric Functions
218(2)
2.4 Limit of a Function
220(23)
2.4.1 Definitions
220(5)
2.4.2 Main Theorems
225(3)
2.4.3 Continuous Functions
228(4)
2.4.4 Several Famous Theorems Related to Continuous Functions
232(3)
2.4.5 Infinite Limits and Limits at Infinities
235(2)
2.4.6 Dealing with Uncertainties
237(2)
2.4.7 Partial Fraction Decomposition Revisited
239(4)
Part II Basics
3 Derivatives
243(70)
3.1 Definition of the Derivative
243(4)
3.2 Main Theorems
247(6)
3.3 Derivatives of Elementary Functions
253(4)
3.4 Complex Numbers Revisited
257(14)
3.4.1 Multiplication and Division of Complex Numbers
257(2)
3.4.2 Moivre Formula
259(1)
3.4.3 Root of a Complex Number
260(2)
3.4.4 Exponential Form. Euler's Formula
262(3)
3.4.5 Solving Cubic Equation
265(3)
3.4.6 Solving Quartic Equation
268(3)
3.5 Approximate Representations of Functions
271(1)
3.6 Differentiation in More Difficult Cases
272(3)
3.7 Higher Order Derivatives
275(8)
3.7.1 Definition and Simple Examples
275(2)
3.7.2 Higher Order Derivatives of Inverse Functions
277(1)
3.7.3 Leibniz Formula
278(3)
3.7.4 Differentiation Operator
281(2)
3.8 Taylor's Formula
283(9)
3.9 Approximate Calculations of Functions
292(2)
3.10 Calculating Limits of Functions in Difficult Cases
294(3)
3.11 Analysing Behaviour of Functions
297(16)
4 Integral
313(98)
4.1 Definite Integral: Introduction
313(6)
4.2 Main Theorems
319(8)
4.3 Main Theorem of Integration. Indefinite Integrals
327(6)
4.4 Indefinite Integrals: Main Techniques
333(26)
4.4.1 Change of Variables
334(3)
4.4.2 Integration by Parts
337(6)
4.4.3 Integration of Rational Functions
343(5)
4.4.4 Integration of Trigonometric Functions
348(3)
4.4.5 Integration of a Rational Function of the Exponential Function
351(1)
4.4.6 Integration of Irrational Functions
352(7)
4.5 More on Calculation of Definite Integrals
359(22)
4.5.1 Change of Variables and Integration by Parts in Definite Integrals
359(3)
4.5.2 Integrals Depending on a Parameter
362(4)
4.5.3 Improper Integrals
366(13)
4.5.4 Cauchy Principal Value
379(2)
4.6 Convolution and Correlation Functions
381(5)
4.7 Applications of Definite Integrals
386(23)
4.7.1 Length of a Curved Line
387(3)
4.7.2 Area of a Plane Figure
390(3)
4.7.3 Volume of Three-Dimensional Bodies
393(4)
4.7.4 A Surface of Revolution
397(2)
4.7.5 Probability Distributions
399(1)
4.7.6 Simple Applications in Physics
400(9)
4.8 Summary
409(2)
5 Functions of Many Variables: Differentiation
411(56)
5.1 Specification of Functions of Many Variables
411(5)
5.2 Limit and Continuity of a Function of Several Variables
416(2)
5.3 Partial Derivatives. Differentiability
418(7)
5.4 A Surface Normal, Tangent Plane
425(2)
5.5 Exact Differentials
427(3)
5.6 Derivatives of Composite Functions
430(11)
5.7 Applications in Thermodynamics
441(4)
5.8 Directional Derivative and the Gradient of a Scalar Field
445(4)
5.9 Taylor's Theorem for Functions of Many Variables
449(4)
5.10 Introduction to Finding an Extremum of a Function
453(14)
5.10.1 Necessary Condition: Stationary Points
453(2)
5.10.2 Characterising Stationary Points: Sufficient Conditions
455(5)
5.10.3 Finding Extrema Subject to Additional Conditions
460(3)
5.10.4 Method of Lagrange Multipliers
463(4)
6 Functions of Many Variables: Integration
467(120)
6.1 Double Integrals
467(21)
6.1.1 Definition and Intuitive Approach
467(2)
6.1.2 Calculation via Iterated Integral
469(8)
6.1.3 Improper Integrals
477(4)
6.1.4 Change of Variables: Jacobian
481(7)
6.2 Volume (Triple) Integrals
488(9)
6.2.1 Definition and Calculation
488(2)
6.2.2 Change of Variables: Jacobian
490(7)
6.3 Applications in Physics: Kinetic Theory of Dilute Gases
497(6)
6.3.1 Maxwell Distribution
497(2)
6.3.2 Gas Equation
499(1)
6.3.3 Kinetic Coefficients
500(3)
6.4 Line Integrals
503(19)
6.4.1 Line Integrals for Scalar Fields
503(5)
6.4.2 Line Integrals for Vector Fields
508(5)
6.4.3 Two-Dimensional Case: Green's Formula
513(4)
6.4.4 Exact Differentials
517(5)
6.5 Surface Integrals
522(31)
6.5.1 Surfaces
523(3)
6.5.2 Area of a Surface
526(5)
6.5.3 Surface Integrals for Scalar Fields
531(2)
6.5.4 Surface Integrals for Vector Fields
533(6)
6.5.5 Relationship Between Line and Surface Integrals Stokes's Theorem
539(7)
6.5.6 Three-Dimensional Case: Exact Differentials
546(2)
6.5.7 Ostrogradsky-Gauss Theorem
548(5)
6.6 Comparison of Line and Surface Integrals
553(1)
6.7 Application of Integral Theorems in Physics. Part I
554(5)
6.7.1 Continuity Equation
554(3)
6.7.2 Archimedes Law
557(2)
6.8 Vector Calculus
559(16)
6.8.1 Divergence of a Vector Field
559(3)
6.8.2 Curl of a Vector Field
562(3)
6.8.3 Vector Fields: Scalar and Vector Potentials
565(10)
6.9 Application of Integral Theorems in Physics. Part II
575(12)
6.9.1 Maxwell's Equations
575(7)
6.9.2 Diffusion and Heat Transport Equations
582(3)
6.9.3 Hydrodynamic Equations of Ideal Liquid (Gas)
585(2)
7 Infinite Numerical and Functional Series
587(70)
7.1 Infinite Numerical Series
588(18)
7.1.1 Series with Positive Terms
590(6)
7.1.2 Multiple Series
596(1)
7.1.3 Euler-Mascheroni Constant
597(1)
7.1.4 Alternating Series
598(3)
7.1.5 General Series: Absolute and Conditional Convergence
601(5)
7.2 Functional Series: General
606(15)
7.2.1 Uniform Convergence
607(2)
7.2.2 Properties: Continuity
609(3)
7.2.3 Properties: Integration and Differentiation
612(2)
7.2.4 Uniform Convergence of Improper Integrals Depending on a Parameter
614(4)
7.2.5 Lattice Sums
618(3)
7.3 Power Series
621(21)
7.3.1 Convergence of the Power Series
622(3)
7.3.2 Uniform Convergence and Term-by-Term Differentiation and Integration of Power Series
625(1)
7.3.3 Taylor Series
626(10)
7.3.4 Bernoulli Numbers and Summation of Powers of Integers
636(1)
7.3.5 Fibonacci Numbers
637(2)
7.3.6 Complex Exponential
639(1)
7.3.7 Taylor Series for Functions of Many Variables
640(2)
7.4 Applications in Physics
642(15)
7.4.1 Diffusion as a Random Walk
642(4)
7.4.2 Coulomb Potential in a Periodic Crystal
646(11)
8 Ordinary Differential Equations
657(104)
8.1 First-Order First Degree Differential Equations
658(26)
8.1.1 Separable Differential Equations
658(4)
8.1.2 "Exact" Differential Equations
662(3)
8.1.3 Method of an Integrating Factor
665(4)
8.1.4 Homogeneous Differential Equations
669(2)
8.1.5 Linear First-Order Differential Equations
671(4)
8.1.6 Examples of Non-linear ODEs
675(2)
8.1.7 Non-linear ODEs: Existence and Uniqueness of Solutions
677(4)
8.1.8 Picard's Method
681(2)
8.1.9 Orthogonal Trajectories
683(1)
8.2 Linear Second-Order Differential Equations
684(23)
8.2.1 General Consideration
684(7)
8.2.2 Homogeneous Linear Differential Equations with Constant Coefficients
691(3)
8.2.3 Nonhomogeneous Linear Differential Equations
694(13)
8.3 Non-linear Second-Order Differential Equations
707(8)
8.3.1 A Few Methods
707(3)
8.3.2 Curve of Pursuit
710(3)
8.3.3 Catenary Curve
713(2)
8.4 Series Solution of Linear ODEs
715(21)
8.4.1 Series Solutions About an Ordinary Point
716(5)
8.4.2 Series Solutions About a Regular Singular Point
721(11)
8.4.3 Special Cases
732(4)
8.5 Linear Systems of Two Differential Equations
736(5)
8.6 Examples in Physics
741(20)
8.6.1 Harmonic Oscillator
741(6)
8.6.2 Falling Water Drop
747(1)
8.6.3 Celestial Mechanics
748(3)
8.6.4 Finite Amplitude Pendulum
751(3)
8.6.5 Tsiolkovsky's Formula
754(2)
8.6.6 Distribution of Particles
756(1)
8.6.7 Residence Probability
757(2)
8.6.8 Defects in a Crystal
759(2)
Index 761
Professor Lev Kantorovich studied theoretical condensed matter physics at the University of Latvia, Riga, Latvia (former part of the USSR), defended his Ph.D. in 1985 in the group of Alex Shluger (currently, at University College London, UK), and then worked at the University of Latvia and the Latvian Medical Academy. From 1993 to 1994, he worked as Visiting Scientist at the University of Oviedo, Spain, and he went on to hold postdoctoral positions at the University of Keele (19946) and University College London (1996-2002), both in the UK. Since 2002, he has worked at Kings College London, initially as Lecturer, then as Reader, and, from 2009, as Professor of Physics. His research interests include the development and application of computational methods for material science, imaging and manipulation at surfaces with atomic probes (AFM and STM), self-assembly of molecules on surfaces, order-N DFT-based methods, quantum conductance with non-equilibrium Greens functions methods, dynamics of open quantum systems using path-integral methods, development and applications of the kinetic Monte Carlo method in growth phenomena, and classical and quantum generalized Langevin equation methods. 
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