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.
Author(s): Lev Kantorovich
Series: Undergraduate Lecture Notes in Physics
Edition: 2
Publisher: Springer
Year: 2022
Language: English
Pages: 791
Tags: Functions; Derivatives; Integral; Series; Ordinary Differential Equations
Introduction to the Second Edition
Introduction to the First Edition
Literature
Famous Scientists Mentioned in the Book
Contents
Part IFundamentals
1 Basic Knowledge
1.1 Logic of Mathematics
1.2 Real Numbers
1.2.1 Integers
1.2.2 Rational Numbers
1.2.3 Irrational Numbers
1.2.4 Real Numbers
1.2.5 Intervals
1.3 Basic Tools: An Introduction
1.3.1 Cartesian Coordinates in 2D and 3D Spaces
1.3.2 Algebra
1.3.3 Inequalities
1.3.4 Functions
1.3.5 Simple Algebraic Equations
1.3.6 Systems of Algebraic Equations
1.3.7 Functional Algebraic Inequalities
1.4 Polynomials
1.4.1 Division of Polynomials
1.4.2 Finding Roots of Polynomials with Integer Coefficients
1.4.3 Vieta's Formulae
1.4.4 Factorisation of Polynomials: Method of Undetermined Coefficients
1.4.5 Multiplication of Polynomials
1.5 Elementary Geometry
1.5.1 Circle, Angles, Lines, Intersections, Polygons
1.5.2 Areas of Simple Plane Figures
1.6 Trigonometric Functions
1.7 Golden Ratio and Golden Triangle. Fibonacci Numbers
1.8 Essential Smooth 2D Curves
1.9 Simple Determinants
1.10 Vectors
1.10.1 Three-Dimensional Space
1.10.2 N-Dimensional Space
1.10.3 My Father's Number Pyramid
1.11 Introduction to Complex Numbers
1.11.1 Cardano's Formula
1.11.2 Complex Numbers
1.11.3 Square Root of a Complex Number
1.11.4 Polynomials with Complex Coefficients
1.11.5 Factorisation of a Polynomial with Real Coefficients
1.12 Summation of Finite Series
1.13 Binomial Formula
1.14 Summae Potestatum and Bernoulli Numbers
1.15 Prime Numbers
1.16 Combinatorics and Multinomial Theorem
1.17 Elements of Classical Probability Theory
1.17.1 Trials, Outcomes and Sets
1.17.2 Definition of Probability of a Random Event
1.17.3 Main Theorems of Probability
1.18 Some Important Inequalities
1.18.1 Cauchy–Bunyakovsky–Schwarz Inequality
1.18.2 Angles Inequality
1.18.3 Four Averages of Positive Numbers
1.19 Lines, Planes and Spheres
1.19.1 Straight Lines
1.19.2 Polar and Spherical Coordinates
1.19.3 Curved Lines
1.19.4 Planes
1.19.5 Circle and Sphere
1.19.6 Typical Problems for Lines, Planes and Spheres
2 Functions
2.1 Definition and Main Types of Functions
2.2 Infinite Numerical Sequences
2.2.1 Definitions
2.2.2 Main Theorems
2.2.3 Sum of an Infinite Numerical Series
2.3 Elementary Functions
2.3.1 Polynomials
2.3.2 Rational Functions
2.3.3 General Power Function
2.3.4 Number e
2.3.5 Exponential Function
2.3.6 Hyperbolic Functions
2.3.7 Logarithmic Function
2.3.8 Trigonometric Functions
2.3.9 Inverse Trigonometric Functions
2.4 Limit of a Function
2.4.1 Definitions
2.4.2 Main Theorems
2.4.3 Continuous Functions
2.4.4 Several Famous Theorems Related to Continuous Functions
2.4.5 Infinite Limits and Limits at Infinities
2.4.6 Dealing with Uncertainties
2.4.7 Partial Fraction Decomposition Revisited
Part IIBasics
3 Derivatives
3.1 Definition of the Derivative
3.2 Main Theorems
3.3 Derivatives of Elementary Functions
3.4 Complex Numbers Revisited
3.4.1 Multiplication and Division of Complex Numbers
3.4.2 Moivre Formula
3.4.3 Root of a Complex Number
3.4.4 Exponential Form. Euler's Formula
3.4.5 Solving Cubic Equation
3.4.6 Solving Quartic Equation
3.5 Approximate Representations of Functions
3.6 Differentiation in More Difficult Cases
3.7 Higher Order Derivatives
3.7.1 Definition and Simple Examples
3.7.2 Higher Order Derivatives of Inverse Functions
3.7.3 Leibniz Formula
3.7.4 Differentiation Operator
3.8 Taylor's Formula
3.9 Approximate Calculations of Functions
3.10 Calculating Limits of Functions in Difficult Cases
3.11 Analysing Behaviour of Functions
4 Integral
4.1 Definite Integral: Introduction
4.2 Main Theorems
4.3 Main Theorem of Integration. Indefinite Integrals
4.4 Indefinite Integrals: Main Techniques
4.4.1 Change of Variables
4.4.2 Integration by Parts
4.4.3 Integration of Rational Functions
4.4.4 Integration of Trigonometric Functions
4.4.5 Integration of a Rational Function of the Exponential Function
4.4.6 Integration of Irrational Functions
4.5 More on Calculation of Definite Integrals
4.5.1 Change of Variables and Integration by Parts in Definite Integrals
4.5.2 Integrals Depending on a Parameter
4.5.3 Improper Integrals
4.5.4 Cauchy Principal Value
4.6 Convolution and Correlation Functions
4.7 Applications of Definite Integrals
4.7.1 Length of a Curved Line
4.7.2 Area of a Plane Figure
4.7.3 Volume of Three-Dimensional Bodies
4.7.4 A Surface of Revolution
4.7.5 Probability Distributions
4.7.6 Simple Applications in Physics
4.8 Summary
5 Functions of Many Variables: Differentiation
5.1 Specification of Functions of Many Variables
5.2 Limit and Continuity of a Function of Several Variables
5.3 Partial Derivatives. Differentiability
5.4 A Surface Normal. Tangent Plane
5.5 Exact Differentials
5.6 Derivatives of Composite Functions
5.7 Applications in Thermodynamics
5.8 Directional Derivative and the Gradient of a Scalar Field
5.9 Taylor's Theorem for Functions of Many Variables
5.10 Introduction to Finding an Extremum of a Function
5.10.1 Necessary Condition: Stationary Points
5.10.2 Characterising Stationary Points: Sufficient Conditions
5.10.3 Finding Extrema Subject to Additional Conditions
5.10.4 Method of Lagrange Multipliers
6 Functions of Many Variables: Integration
6.1 Double Integrals
6.1.1 Definition and Intuitive Approach
6.1.2 Calculation via Iterated Integral
6.1.3 Improper Integrals
6.1.4 Change of Variables: Jacobian
6.2 Volume (Triple) Integrals
6.2.1 Definition and Calculation
6.2.2 Change of Variables: Jacobian
6.3 Applications in Physics: Kinetic Theory of Dilute Gases
6.3.1 Maxwell Distribution
6.3.2 Gas Equation
6.3.3 Kinetic Coefficients
6.4 Line Integrals
6.4.1 Line Integrals for Scalar Fields
6.4.2 Line Integrals for Vector Fields
6.4.3 Two-Dimensional Case: Green's Formula
6.4.4 Exact Differentials
6.5 Surface Integrals
6.5.1 Surfaces
6.5.2 Area of a Surface
6.5.3 Surface Integrals for Scalar Fields
6.5.4 Surface Integrals for Vector Fields
6.5.5 Relationship Between Line and Surface Integrals. Stokes's Theorem
6.5.6 Three-Dimensional Case: Exact Differentials
6.5.7 Ostrogradsky–Gauss Theorem
6.6 Comparison of Line and Surface Integrals
6.7 Application of Integral Theorems in Physics. Part I
6.7.1 Continuity Equation
6.7.2 Archimedes Law
6.8 Vector Calculus
6.8.1 Divergence of a Vector Field
6.8.2 Curl of a Vector Field
6.8.3 Vector Fields: Scalar and Vector Potentials
6.9 Application of Integral Theorems in Physics. Part II
6.9.1 Maxwell's Equations
6.9.2 Diffusion and Heat Transport Equations
6.9.3 Hydrodynamic Equations of Ideal Liquid (Gas)
7 Infinite Numerical and Functional Series
7.1 Infinite Numerical Series
7.1.1 Series with Positive Terms
7.1.2 Multiple Series
7.1.3 Euler-Mascheroni Constant
7.1.4 Alternating Series
7.1.5 General Series: Absolute and Conditional Convergence
7.2 Functional Series: General
7.2.1 Uniform Convergence
7.2.2 Properties: Continuity
7.2.3 Properties: Integration and Differentiation
7.2.4 Uniform Convergence of Improper Integrals Depending on a Parameter
7.2.5 Lattice Sums
7.3 Power Series
7.3.1 Convergence of the Power Series
7.3.2 Uniform Convergence and Term-by-Term Differentiation and Integration of Power Series
7.3.3 Taylor Series
7.3.4 Bernoulli Numbers and Summation of Powers of Integers
7.3.5 Fibonacci Numbers
7.3.6 Complex Exponential
7.3.7 Taylor Series for Functions of Many Variables
7.4 Applications in Physics
7.4.1 Diffusion as a Random Walk
7.4.2 Coulomb Potential in a Periodic Crystal
8 Ordinary Differential Equations
8.1 First-Order First Degree Differential Equations
8.1.1 Separable Differential Equations
8.1.2 ``Exact'' Differential Equations
8.1.3 Method of an Integrating Factor
8.1.4 Homogeneous Differential Equations
8.1.5 Linear First-Order Differential Equations
8.1.6 Examples of Non-linear ODEs
8.1.7 Non-linear ODEs: Existence and Uniqueness of Solutions
8.1.8 Picard's Method
8.1.9 Orthogonal Trajectories
8.2 Linear Second-Order Differential Equations
8.2.1 General Consideration
8.2.2 Homogeneous Linear Differential Equations with Constant Coefficients
8.2.3 Nonhomogeneous Linear Differential Equations
8.3 Non-linear Second-Order Differential Equations
8.3.1 A Few Methods
8.3.2 Curve of Pursuit
8.3.3 Catenary Curve
8.4 Series Solution of Linear ODEs
8.4.1 Series Solutions About an Ordinary Point
8.4.2 Series Solutions About a Regular Singular Point
8.4.3 Special Cases
8.5 Linear Systems of Two Differential Equations
8.6 Examples in Physics
8.6.1 Harmonic Oscillator
8.6.2 Falling Water Drop
8.6.3 Celestial Mechanics
8.6.4 Finite Amplitude Pendulum
8.6.5 Tsiolkovsky's Formula
8.6.6 Distribution of Particles
8.6.7 Residence Probability
8.6.8 Defects in a Crystal
Index
Index
