Math Tools: 500+ Applications in Science and Arts

Table of Contents ... 5
1 Introduction ... 7
2 Equations, systems of equations ... 13
	2.1 The fundamentals of numbers and equations ... 14
	2.2 Linear equations ... 29
	2.3 Systems of linear equations ... 33
	2.4 Quadratic equations ... 42
	2.5 Algebraic equations of higher degree ... 47
	2.6 Further applications ... 52
3 Proportions and similar objects ... 73
	3.1 Similarity of planar figures ... 74
	3.2 Similarity of spatial objects ... 80
	3.3 On small scales not as on large scales ... 84
	3.4 Centrifugal and gravitational forces ... 97
	3.5 Further applications ... 103
4 Angles and trigonometry ... 125
	4.1 The family of Pythagorean theorems ... 126
	4.2 Radian measure ... 131
	4.3 Sine, cosine, tangent ... 137
	4.4 The scalene triangle ... 151
	4.5 Further applications ... 160
5 Vector analysis ... 181
	5.1 Elementary vector operations ... 182
	5.2 Dot product and cross product ... 194
	5.3 Intersecting lines and planes ... 199
	5.4 Distances, angles, areas, volumes ... 203
	5.5 Reflection ... 213
	5.6 Further Applications ... 222
6 Functions and their derivatives ... 243
	6.1 Real functions and their inverses ... 244
	6.2 Power, exponential, and logarithmic functions ... 250
	6.3 The derivative function of a real function ... 264
	6.4 Differentiation rules ... 269
	6.5 Differentiating with a computer ... 285
	6.6 Solving equations of the form f(x) = 0 ... 287
	6.7 Further applications ... 293
7 Curves and surfaces ... 303
	7.1 Rigid body motions ... 304
	7.2 Matrix calculations and some applications ... 318
	7.3 Parametrization of curves ... 321
	7.4 Envelopes ... 347
	7.5 Surfaces ... 356
	7.6 Further applications ... 364
8 Infinitesimal calculus ... 373
	8.1 Calculation with infinitesimal quantities ... 374
	8.2 Curve sketching ... 377
	8.3 Optimization problems ... 380
	8.4 Series expansion ... 387
	8.5 Integration as the inverse of differentiation ... 395
	8.6 Interpretations of the definite integrals ... 401
	8.7 Numerical integration ... 419
	8.8 Further applications ... 426
9 Statistics and probability calculus ... 443
	9.1 Descriptive statistics ... 444
	9.2 Probability – computing with chance ... 452
	9.3 The probability concept ... 459
	9.4 Conditional and independent events ... 465
	9.5 Combinatorics ... 473
	9.6 Fallacies, traps of reasoning, and apparent contradictions ... 481
	9.7 Probability distributions ... 488
	9.8 Further applications ... 497
A Music and mathematics ... 507
	A.1 Basic approach, fundamentals of natural science ... 508
	A.2 System formation ... 511
	A.3 Tuning instruments – intonation ... 513
	A.4 Numerical symbolism ... 518
	A.5 Harmonics (basic fundamental research) ... 519
	A.6 Numerical examples ... 521
B Numbers ... 525
	B.1 Numerology ... 526
	B.2 Rational and irrational numbers ... 530
	B.3 Famous irrational numbers ... 533
	B.4 The Fibonacci numbers ... 536
	B.5 Imaginary and complex numbers ... 539
Index ... 559