Handbook of mathematics for engineers and scienteists part 1 ppsx

HANDBOOK OF
MATHEMATICS FOR
ENGINEERS AND
SCIENTISTS
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HANDBOOK OF
MATHEMATICS FOR
ENGINEERS AND
SCIENTISTS
Andrei D. Polyanin
Alexander V. Manzhirov
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Chapman & Hall/CRC
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CONTENTS
Authors xxv
Preface xxvii
Main Notation xxix
Part I. Definitions, Formulas, Methods, and Theorems 1
1. Arithmetic and Elementary Algebra 3
1.1. RealNumbers 3
1.1.1. IntegerNumbers 3
1.1.2. Real, Rational, and Irrational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2. Equalities and Inequalities. Arithmetic Operations. Absolute Value . . . . . . . . . . . . . . . . 5
1.2.1. Equalities and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2. Addition and Multiplication of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3. Ratios and Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.4. Percentage 7
1.2.5. Absolute Value of a Number (Modulus of a Number) . . . . . . . . . . . . . . . . . . . . . . 8
1.3. PowersandLogarithms 8
1.3.1. PowersandRoots 8
1.3.2. Logarithms 9

1.4. BinomialTheoremandRelatedFormulas 10
1.4.1. Factorials. Binomial Coefficients.BinomialTheorem 10
1.4.2. RelatedFormulas 10
1.5. Arithmetic and Geometric Progressions. Finite Sums and Products . . . . . . . . . . . . . . . . . 11
1.5.1. ArithmeticandGeometricProgressions 11
1.5.2. Finite Series and Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6. Mean Values and Inequalities of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6.1. Arithmetic Mean, Geometric Mean, and Other Mean Values. Inequalities for
MeanValues 13
1.6.2. Inequalities of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.7. Some Mathematical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.7.1. ProofbyContradiction 15
1.7.2. Mathematical Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7.3. Proof by Counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.7.4. Method of Undetermined Coefficients 17
ReferencesforChapter1 18
2. Elementary Functions 19
2.1. Power, Exponential, and Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.1. Power Function: y = x
α
19
2.1.2. Exponential Function: y = a
x
21
2.1.3. Logarithmic Function: y =log
a
x 22
2.2. Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.1. Trigonometric Circle. Definition of Trigonometric Functions . . . . . . . . . . . . . . . 24
2.2.2. Graphs of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.3. Properties of Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
v
vi CONTENTS
2.3. Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.1. Definitions. Graphs of Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . 30
2.3.2. Properties of Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4. HyperbolicFunctions 34
2.4.1. Definitions. Graphs of Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.2. Properties of Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5. InverseHyperbolicFunctions 39
2.5.1. Definitions. Graphs of Inverse Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . 39
2.5.2. Properties of Inverse Hyperbolic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
ReferencesforChapter2 42
3. Elementary Geometry 43
3.1. PlaneGeometry 43
3.1.1. Triangles 43
3.1.2. Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.3. Circle 56
3.2. SolidGeometry 59
3.2.1. StraightLines,Planes,andAnglesinSpace 59
3.2.2. Polyhedra 61
3.2.3. SolidsFormedbyRevolutionofLines 65
3.3. Spherical Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.3.1. SphericalGeometry 70
3.3.2. SphericalTriangles 71
ReferencesforChapter3 75
4. Analytic Geometry 77
4.1. Points,Segments,andCoordinatesonLineandPlane 77
4.1.1. CoordinatesonLine 77
4.1.2. CoordinatesonPlane 78

4.1.3. PointsandSegmentsonPlane 81
4.2. CurvesonPlane 84
4.2.1. CurvesandTheirEquations 84
4.2.2. MainProblemsofAnalyticGeometryforCurves 88
4.3. StraightLinesandPointsonPlane 89
4.3.1. EquationsofStraightLinesonPlane 89
4.3.2. MutualArrangementofPointsandStraightLines 93
4.4. Second-Order Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.4.1. Circle 97
4.4.2. Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.4.3. Hyperbola 101
4.4.4. Parabola 104
4.4.5. Transformation of Second-Order Curves to Canonical Form . . . . . . . . . . . . . . . . 107
4.5. Coordinates, Vectors, Curves, and Surfaces in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.5.1. Vectors.CartesianCoordinateSystem 113
4.5.2. CoordinateSystems 114
4.5.3. Vectors. Products of Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.5.4. Curves and Surfaces in Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123