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