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E-grāmata: Mathematics for Natural Scientists: Fundamentals and Basics

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This book 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, avoiding precise mathematical jargon and proofs which are comprehensible only to mathematicians. Instead, all proofs are given in a form that is clear and convincing enough for a physicist. Examples, where appropriate, are given from physics contexts. Both solved and unsolved problems are provided in each section of the book.

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.

Part I Fundamentals
1 Basic Knowledge
3(64)
1.1 Logic of Mathematics
3(2)
1.2 Real Numbers
5(5)
1.3 Cartesian Coordinates in 2D and 3D Spaces
10(1)
1.4 Elementary Geometry
11(3)
1.5 Introduction to Elementary Functions and Trigonometry
14(9)
1.6 Simple Determinants
23(3)
1.7 Vectors
26(11)
1.7.1 Three-Dimensional Space
26(9)
1.7.2 N-dimensional Space
35(2)
1.8 Introduction to Complex Numbers
37(4)
1.9 Summation of Finite Series
41(3)
1.10 Binomial Formula
44(5)
1.11 Combinatorics and Multinomial Theorem
49(3)
1.12 Some Important Inequalities
52(4)
1.13 Lines and Planes
56(11)
1.13.1 Straight Line
56(1)
1.13.2 Polar and Spherical Coordinates
57(1)
1.13.3 Curved Lines
58(1)
1.13.4 Planes
59(3)
1.13.5 Typical Problems for Lines and Planes
62(5)
2 Functions
67(56)
2.1 Definition and Main Types of Functions
67(4)
2.2 Infinite Numerical Sequences
71(7)
2.2.1 Definitions
71(2)
2.2.2 Main Theorems
73(4)
2.2.3 Sum of an Infinite Numerical Series
77(1)
2.3 Elementary Functions
78(22)
2.3.1 Polynomials
79(1)
2.3.2 Rational Functions
80(4)
2.3.3 General Power Function
84(2)
2.3.4 Number e
86(4)
2.3.5 Exponential Function
90(1)
2.3.6 Hyperbolic Functions
91(1)
2.3.7 Logarithmic Function
91(2)
2.3.8 Trigonometric Functions
93(5)
2.3.9 Inverse Trigonometric Functions
98(2)
2.4 Limit of a Function
100(23)
2.4.1 Definitions
100(5)
2.4.2 Main Theorems
105(3)
2.4.3 Continuous Functions
108(4)
2.4.4 Several Famous Theorems Related to Continuous Functions
112(3)
2.4.5 Infinite Limits and Limits at Infinities
115(2)
2.4.6 Dealing with Uncertainties
117(6)
Part II Basics
3 Derivatives
123(52)
3.1 Definition of the Derivative
123(4)
3.2 Main Theorems
127(5)
3.3 Derivatives of Elementary Functions
132(4)
3.4 Approximate Representations of Functions
136(1)
3.5 Differentiation in More Difficult Cases
137(3)
3.6 Higher Order Derivatives
140(6)
3.7 Taylor's Formula
146(9)
3.8 Approximate Calculations of Functions
155(2)
3.9 Calculating Limits of Functions in Difficult Cases
157(3)
3.10 Analysing Behaviour of Functions
160(15)
4 Integral
175(86)
4.1 Definite Integral: Introduction
175(6)
4.2 Main Theorems
181(7)
4.3 Main Theorem of Integration: Indefinite Integrals
188(7)
4.4 Indefinite Integrals: Main Techniques
195(25)
4.4.1 Change of Variables
195(3)
4.4.2 Integration by Parts
198(6)
4.4.3 Integration of Rational Functions
204(5)
4.4.4 Integration of Trigonometric Functions
209(3)
4.4.5 Integration of a Rational Function of the Exponential Function
212(1)
4.4.6 Integration of Irrational Functions
213(7)
4.5 More on Calculation of Definite Integrals
220(17)
4.5.1 Change of Variables and Integration by Parts in Definite Integrals
220(3)
4.5.2 Integrals Depending on a Parameter
223(3)
4.5.3 Improper Integrals
226(9)
4.5.4 Cauchy Principal Value
235(2)
4.6 Applications of Definite Integrals
237(22)
4.6.1 Length of a Curved Line
238(4)
4.6.2 Area of a Plane Figure
242(3)
4.6.3 Volume of Three-Dimensional Bodies
245(3)
4.6.4 A Surface of Revolution
248(2)
4.6.5 Simple Applications in Physics
250(9)
4.7 Summary
259(2)
5 Functions of Many Variables: Differentiation
261(54)
5.1 Specification of Functions of Many Variables
261(5)
5.1.1 Sphere
262(1)
5.1.2 Ellipsoid
262(2)
5.1.3 One-Pole (One Sheet) Hyperboloid
264(1)
5.1.4 Two-Pole (Two Sheet) Hyperboloid
264(1)
5.1.5 Hyperbolic Paraboloid
264(2)
5.2 Limit and Continuity of a Function of Several Variables
266(2)
5.3 Partial Derivatives: Differentiability
268(7)
5.4 A Surface Normal. Tangent Plane
275(2)
5.5 Exact Differentials
277(3)
5.6 Derivatives of Composite Functions
280(10)
5.7 Applications in Thermodynamics
290(4)
5.8 Directional Derivative and the Gradient of a Scalar Field
294(5)
5.9 Taylor's Theorem for Functions of Many Variables
299(2)
5.10 Introduction to Finding an Extremum of a Function
301(14)
5.10.1 Necessary Condition: Stationary Points
302(2)
5.10.2 Characterising Stationary Points: Sufficient Conditions
304(4)
5.10.3 Finding Extrema Subject to Additional Conditions
308(2)
5.10.4 Method of Lagrange Multipliers
310(5)
6 Functions of Many Variables: Integration
315(102)
6.1 Double Integrals
315(18)
6.1.1 Definition and Intuitive Approach
315(2)
6.1.2 Calculation via Iterated Integral
317(6)
6.1.3 Improper Integrals
323(4)
6.1.4 Change of Variables: Jacobian
327(6)
6.2 Volume (Triple) Integrals
333(5)
6.2.1 Definition and Calculation
333(2)
6.2.2 Change of Variables: Jacobian
335(3)
6.3 Line Integrals
338(17)
6.3.1 Line Integrals for Scalar Fields
338(4)
6.3.2 Line Integrals for Vector Fields
342(4)
6.3.3 Two-Dimensional Case: Green's Formula
346(5)
6.3.4 Exact Differentials
351(4)
6.4 Surface Integrals
355(29)
6.4.1 Surfaces
355(5)
6.4.2 Area of a Surface
360(4)
6.4.3 Surface Integrals for Scalar Fields
364(2)
6.4.4 Surface Integrals for Vector Fields
366(5)
6.4.5 Relationship Between Line and Surface Integrals: Stokes's Theorem
371(8)
6.4.6 Three-Dimensional Case: Exact Differentials
379(2)
6.4.7 Ostrogradsky--Gauss Theorem
381(3)
6.5 Application of Integral Theorems in Physics: Part I
384(4)
6.5.1 Continuity Equation
384(3)
6.5.2 Archimedes Law
387(1)
6.6 Vector Calculus
388(16)
6.6.1 Divergence of a Vector Field
388(3)
6.6.2 Curl of a Vector Field
391(3)
6.6.3 Vector Fields: Scalar and Vector Potentials
394(10)
6.7 Application of Integral Theorems in Physics: Part II
404(13)
6.7.1 Maxwell's Equations
404(7)
6.7.2 Diffusion and Heat Transport Equations
411(2)
6.7.3 Hydrodynamic Equations of Ideal Liquid (Gas)
413(4)
7 Infinite Numerical and Functional Series
417(38)
7.1 Infinite Numerical Series
418(16)
7.1.1 Series with Positive Terms
420(5)
7.1.2 Euler--Mascheroni Constant
425(1)
7.1.3 Alternating Series
426(3)
7.1.4 General Series: Absolute and Conditional Convergence
429(5)
7.2 Functional Series: General
434(7)
7.2.1 Uniform Convergence
435(2)
7.2.2 Properties: Continuity
437(2)
7.2.3 Properties: Integration and Differentiation
439(2)
7.3 Power Series
441(14)
7.3.1 Convergence of the Power Series
442(3)
7.3.2 Uniform Convergence and Term-by-Term Differentiation and Integration of Power Series
445(1)
7.3.3 Taylor Series
446(9)
8 Ordinary Differential Equations
455(66)
8.1 First Order First Degree Differential Equations
456(12)
8.1.1 Separable Differential Equations
456(2)
8.1.2 "Exact" Differential Equations
458(2)
8.1.3 Method of an Integrating Factor
460(2)
8.1.4 Homogeneous Differential Equations
462(2)
8.1.5 Linear First Order Differential Equations
464(4)
8.2 Linear Second Order Differential Equations
468(15)
8.2.1 Homogeneous Linear Differential Equations with Constant Coefficients
471(3)
8.2.2 Inhomogeneous Linear Differential Equations
474(9)
8.3 Non-linear Second Order Differential Equations
483(3)
8.4 Series Solution of Linear ODEs
486(20)
8.4.1 Series Solutions About an Ordinary Point
487(4)
8.4.2 Series Solutions About a Regular Singular Point
491(10)
8.4.3 Special Cases
501(5)
8.5 Examples in Physics
506(15)
8.5.1 Harmonic Oscillator
506(7)
8.5.2 Falling Water Drop
513(1)
8.5.3 Tsiolkovsky's Formula
514(1)
8.5.4 Distribution of Particles
515(2)
8.5.5 Residence Probability
517(1)
8.5.6 Defects in a Crystal
518(3)
Index 521
Lev Kantorovich is a member of the Physics faculty at King's College London. He has published two books and over 190 peer reviewed papers. Prof. Kantorovich has taught mathematical methods in physics at King's College for the past 12 years, receiving two Teaching Excellence Awards.
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