A Concise Handbook of Mathematics, Physics, and Engineering Sciences

1127 Pages·2010·20.23 MB·English

by  Andrei D. Polyanin

A CONCISE HANDBOOK OF
MATHEMATICS,
PHYSICS, AND
ENGINEERING
SCIENCES
K10319_FM.indd 1 9/20/10 4:27:31 PM
A CONCISE HANDBOOK OF
MATHEMATICS,
PHYSICS, AND
ENGINEERING
SCIENCES
Edited by
Andrei D. Polyanin

Alexei I. Chernoutsan
Authors
A.I. Chernoutsan, A.V. Egorov, A.V. Manzhirov, A.D. Polyanin,
V.D. Polyanin, V.A. Popov, B.V. Putyatin, Yu.V. Repina,
V.M. Safrai, A.I. Zhurov
K10319_FM.indd 2 9/20/10 4:27:32 PM
A CONCISE HANDBOOK OF
MATHEMATICS,
PHYSICS, AND
ENGINEERING
SCIENCES
Edited by
Andrei D. Polyanin

Alexei I. Chernoutsan
Authors
A.I. Chernoutsan, A.V. Egorov, A.V. Manzhirov, A.D. Polyanin,
V.D. Polyanin, V.A. Popov, B.V. Putyatin, Yu.V. Repina,
V.M. Safrai, A.I. Zhurov
K10319_FM.indd 3 9/20/10 4:27:32 PM
CRC Press
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K10319:A.D.PolyaninandA.I.Chernoutsan,Aconcisehandbookofmathematics,physics,andengineeringsciences,Chapman&Hall/CRC,2010(v.6,Aug2010) Pagev
CONTENTS
Preface .................................................................. xxv
Editors .................................................................. xxvii
PartI. Mathematics 1
M1. ArithmeticandElementaryAlgebra .................................. 3
M1.1. RealNumbers ...................................................... 3
M1.1.1. IntegerNumbers ............................................ 3
M1.1.2. Real,Rational, andIrrational Numbers ......................... 4
M1.2. EqualitiesandInequalities. ArithmeticOperations. AbsoluteValue ........ 5
M1.2.1. EqualitiesandInequalities .................................... 5
M1.2.2. AdditionandMultiplication ofNumbers ....................... 5
M1.2.3. RatiosandProportions ....................................... 6
M1.2.4. Percentage ................................................. 6
M1.2.5. AbsoluteValueofaNumber(ModulusofaNumber) ............ 7
M1.3. PowersandLogarithms .............................................. 7
M1.3.1. PowersandRoots ........................................... 7
M1.3.2. Logarithms ................................................. 8
M1.4. BinomialTheoremandRelatedFormulas ............................... 9
M1.4.1. Factorials. BinomialCoefficients. BinomialTheorem ............ 9
M1.4.2. RelatedFormulas ........................................... 9
M1.5. Progressions ........................................................ 10
M1.5.1. ArithmeticProgression ....................................... 10
M1.5.2. GeometricProgression ....................................... 10
M1.6. MeanValuesandSomeInequalities .................................... 11
M1.6.1. ArithmeticMean,GeometricMean,andOtherMeanValues ...... 11
M1.6.2. Inequalities forMeanValues .................................. 11
M1.6.3. SomeInequalities ofGeneralForm ............................ 11
M1.7. SomeMathematicalMethods ......................................... 12
M1.7.1. ProofbyContradiction ....................................... 12
M1.7.2. Mathematical Induction ...................................... 12
M1.7.3. ProofbyCounterexample .................................... 13
Bibliography forChapterM1 ............................................... 13
M2. ElementaryFunctions ............................................... 15
M2.1. Power,Exponential, andLogarithmicFunctions ......................... 15
M2.1.1. PowerFunction: y = xα ...................................... 15
M2.1.2. Exponential Function: y =ax ................................. 17
M2.1.3. LogarithmicFunction: y = log x ............................. 18
a
M2.2. Trigonometric Functions ............................................. 19
M2.2.1. Trigonometric Circle. DefinitionofTrigonometric Functions ...... 19
M2.2.2. GraphsofTrigonometricFunctions ............................ 21
M2.2.3. PropertiesofTrigonometric Functions ......................... 22
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vi CONTENTS
M2.3. InverseTrigonometric Functions ...................................... 26
M2.3.1. Definitions. GraphsofInverse TrigonometricFunctions .......... 26
M2.3.2. PropertiesofInverseTrigonometric Functions ................... 27
M2.4. Hyperbolic Functions ................................................ 29
M2.4.1. Definitions. GraphsofHyperbolic Functions .................... 29
M2.4.2. PropertiesofHyperbolic Functions ............................ 30
M2.5. InverseHyperbolic Functions ......................................... 33
M2.5.1. Definitions. GraphsofInverse HyperbolicFunctions ............. 33
M2.5.2. PropertiesofInverseHyperbolic Functions ..................... 34
Bibliography forChapterM2 ............................................... 35
M3. ElementaryGeometry ............................................... 37
M3.1. PlaneGeometry ..................................................... 37
M3.1.1. Triangles ................................................... 37
M3.1.2. Polygons ................................................... 43
M3.1.3. Circle ...................................................... 46
M3.2. SolidGeometry ..................................................... 49
M3.2.1. StraightLines,Planes,andAnglesinSpace ..................... 49
M3.2.2. Polyhedra .................................................. 51
M3.2.3. SolidsFormedbyRevolution ofLines ......................... 55
Bibliography forChapterM3 ............................................... 60
M4. AnalyticGeometry .................................................. 61
M4.1. Points,Segments,andCoordinate Plane ................................ 61
M4.1.1. CartesianandPolarCoordinates onPlane ....................... 61
M4.1.2. DistanceBetweenPoints. DivisionofSegmentinGivenRatio. Area
ofaPolygon ................................................ 62
M4.2. StraightLinesandPointsonPlane ..................................... 64
M4.2.1. EquationsofStraightLinesonPlane ........................... 64
M4.2.2. MutualArrangement ofPointsandStraightLines ................ 66
M4.3. QuadraticCurves .................................................... 68
M4.3.1. Circle ...................................................... 68
M4.3.2. Ellipse ..................................................... 69
M4.3.3. Hyperbola .................................................. 71
M4.3.4. Parabola ................................................... 74
M4.3.5. Transformation ofQuadraticCurvestoCanonicalForm .......... 75
M4.4. Coordinates, Vectors,Curves,andSurfacesinSpace ..................... 78
M4.4.1. VectorsandTheirProperties .................................. 78
M4.4.2. Coordinate Systems ......................................... 79
M4.4.3. Scalar,Cross,andScalarTripleProductsofVectors .............. 82
M4.5. LineandPlaneinSpace .............................................. 84
M4.5.1. PlaneinSpace .............................................. 84
M4.5.2. LineinSpace ............................................... 89
M4.5.3. MutualArrangement ofPoints,Lines,andPlanes ................ 91
M4.6. QuadricSurfaces(Quadrics) .......................................... 96
M4.6.1. QuadricsandTheirCanonicalEquations ....................... 96
M4.6.2. Quadrics(GeneralTheory) ................................... 99
Bibliography forChapterM4 ............................................... 102
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CONTENTS vii
M5. Algebra ............................................................. 103
M5.1. PolynomialsandAlgebraicEquations .................................. 103
M5.1.1. PolynomialsandTheirProperties .............................. 103
M5.1.2. LinearandQuadraticEquations ............................... 105
M5.1.3. CubicEquations ............................................ 105
M5.1.4. Fourth DegreeEquation ...................................... 107
M5.1.5. AlgebraicEquations ofArbitraryDegreeandTheirProperties ..... 108
M5.2. DeterminantsandMatrices ........................................... 112
M5.2.1. Determinants ............................................... 112
M5.2.2. Matrices. TypesofMatrices. Operations withMatrices ........... 114
M5.2.3. InverseMatrix. Functions ofMatrices .......................... 119
M5.2.4. Eigenvalues and Characteristic Equation of a Matrix. The
Cayley–Hamilton Theorem ................................... 120
M5.3. SystemsofLinearAlgebraicEquations ................................. 123
M5.3.1. Consistency ConditionforaLinearSystem ..................... 123
M5.3.2. FindingSolutionsofaSystemofLinearEquations .............. 124
M5.4. QuadraticForms .................................................... 127
M5.4.1. QuadraticFormsandTheirTransformations .................... 127
M5.4.2. CanonicalandNormalRepresentations ofaQuadraticForm ...... 129
M5.5. LinearSpaces ....................................................... 131
M5.5.1. ConceptofaLinearSpace. ItsBasisandDimension ............. 131
M5.5.2. SubspacesofLinearSpaces ................................... 134
M5.5.3. CoordinateTransformationsCorrespondingtoBasisTransformations
inaLinearSpace ............................................ 135
M5.5.4. EuclideanSpace ............................................ 136
Bibliography forChapterM5 ............................................... 138
M6. LimitsandDerivatives ............................................... 139
M6.1. BasicConceptsofMathematical Analysis .............................. 139
M6.1.1. NumberSets. FunctionsofRealVariable ....................... 139
M6.1.2. LimitofaSequence ......................................... 141
M6.1.3. LimitofaFunction. Asymptotes .............................. 144
M6.1.4. InfinitelySmallandInfinitelyLargeFunctions .................. 146
M6.1.5. Continuous Functions. Discontinuities of the First and the Second
Kind ....................................................... 147
M6.1.6. ConvexandConcaveFunctions ............................... 148
M6.1.7. Convergence ofFunctions .................................... 149
M6.2. DifferentialCalculusforFunctionsofaSingleVariable .................. 150
M6.2.1. Derivative and Differential: Their Geometrical and Physical
Meaning ................................................... 150
M6.2.2. TableofDerivativesandDifferentiation Rules ................... 151
M6.2.3. TheoremsaboutDifferentiable Functions. L’HospitalRule ........ 153
M6.2.4. Higher Order DerivativesandDifferentials. Taylor’sFormula ..... 154
M6.2.5. ExtremalPoints. PointsofInflection ........................... 156
M6.2.6. QualitativeAnalysisofFunctions andConstruction ofGraphs ..... 158
M6.2.7. Approximate Solution of Equations (Root Finding Algorithms for
Continuous Functions) ....................................... 160
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viii CONTENTS
M6.3. FunctionsofSeveralVariables. PartialDerivatives ....................... 162
M6.3.1. PointSets. Functions. LimitsandContinuity ................... 162
M6.3.2. Differentiation ofFunctions ofSeveralVariables ................ 163
M6.3.3. Directional Derivative. Gradient. Geometrical Applications ....... 166
M6.3.4. ExtremalPointsofFunctionsofSeveralVariables ............... 167
M6.3.5. Differential OperatorsoftheFieldTheory ...................... 170
Bibliography forChapterM6 ............................................... 170
M7. Integrals ............................................................ 171
M7.1. IndefiniteIntegral ................................................... 171
M7.1.1. Antiderivative. IndefiniteIntegralandItsProperties .............. 171
M7.1.2. Table of Basic Integrals. Properties of the Indefinite Integral.
ExamplesofIntegration ...................................... 172
M7.1.3. Integration ofRationalFunctions .............................. 174
M7.1.4. Integration ofIrrational Functions ............................. 177
M7.1.5. Integration ofExponential andTrigonometric Functions .......... 179
M7.1.6. Integration ofPolynomialsMultiplied byElementaryFunctions ... 181
M7.2. DefiniteIntegral ..................................................... 183
M7.2.1. Basic Definitions. Classes of Integrable Functions. Geometrical
MeaningoftheDefiniteIntegral ............................... 183
M7.2.2. PropertiesofDefiniteIntegrals andUsefulFormulas ............. 184
M7.2.3. AsymptoticFormulasfortheCalculation ofIntegrals ............ 185
M7.2.4. Mean Value Theorems. Properties of Integrals in Terms of
Inequalities ................................................. 187
M7.2.5. GeometricandPhysicalApplications oftheDefiniteIntegral ...... 188
M7.2.6. ImproperIntegralswithInfiniteIntegration Limits ............... 190
M7.2.7. ImproperIntegralsofUnbounded Functions .................... 193
M7.2.8. Approximate (Numerical) Methods for Computation of Definite
Integrals ................................................... 194
M7.3. DoubleandTripleIntegrals ........................................... 195
M7.3.1. DefinitionandProperties oftheDoubleIntegral ................. 195
M7.3.2. Computation oftheDoubleIntegral ............................ 197
M7.3.3. GeometricandPhysicalApplications oftheDoubleIntegral ...... 199
M7.3.4. DefinitionandProperties oftheTripleIntegral .................. 200
M7.3.5. Computation of the Triple Integral. Some Applications. Iterated
IntegralsandAsymptoticFormulas ............................ 201
M7.4. LineandSurfaceIntegrals ............................................ 203
M7.4.1. LineIntegraloftheFirstKind ................................. 203
M7.4.2. LineIntegraloftheSecondKind .............................. 205
M7.4.3. SurfaceIntegral oftheFirstKind .............................. 206
M7.4.4. SurfaceIntegral oftheSecondKind ........................... 207
M7.4.5. IntegralFormulasofVectorCalculus ........................... 208
Bibliography forChapterM7 ............................................... 209
M8. Series ............................................................... 211
M8.1. NumericalSeriesandInfiniteProducts ................................. 211
M8.1.1. Convergent Numerical Series and Their Properties. Cauchy’s
Criterion ................................................... 211
M8.1.2. Convergence CriteriaforSerieswithPositive(Nonnegative) Terms 212
M8.1.3. Convergence Criteria for Arbitrary Numerical Series. Absolute and
Conditional Convergence ..................................... 213
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CONTENTS ix
M8.1.4. Multiplication ofSeries. SomeInequalities ..................... 215
M8.2. FunctionSeries ..................................................... 215
M8.2.1. PointwiseandUniformConvergence ofFunctionSeries .......... 215
M8.2.2. Basic Criteria of Uniform Convergence. Properties of Uniformly
Convergent Series ........................................... 216
M8.3. PowerSeries ........................................................ 218
M8.3.1. RadiusofConvergence ofPowerSeries. Properties ofPowerSeries 218
M8.3.2. TaylorandMaclaurinPowerSeries ............................ 220
M8.3.3. Operations with Power Series. Summation Formulas for Power
Series ...................................................... 221
M8.4. FourierSeries ....................................................... 223
M8.4.1. Representation of 2π Periodic Functions by Fourier Series. Main
Results ..................................................... 223
M8.4.2. Fourier Expansions of Periodic, Nonperiodic, Even, and Odd
Functions .................................................. 225
M8.4.3. CriteriaofUniformandMean SquareConvergenceofFourierSeries 226
Bibliography forChapterM8 ............................................... 227
M9. FunctionsofComplexVariable ....................................... 229
M9.1. ComplexNumbers .................................................. 229
M9.1.1. Definition of a Complex Number. Arithmetic Operations with
ComplexNumbers .......................................... 229
M9.1.2. Trigonometric FormofComplexNumbers. PowersandRadicals .. 230
M9.2. FunctionsofComplexVariables ....................................... 231
M9.2.1. BasicConcepts. DifferentiationofaFunctionofaComplexVariable 231
M9.2.2. Integration ofFunctionsofComplexVariables .................. 238
M9.2.3. TaylorandLaurentSeries .................................... 241
M9.2.4. ZerosandIsolatedSingularities ofAnalyticFunctions ............ 243
M9.2.5. Residues. Calculation ofDefiniteIntegrals ..................... 245
Bibliography forChapterM9 ............................................... 249
M10. IntegralTransforms ................................................ 251
M10.1. GeneralFormofIntegralTransforms. Inversion Formulas ............... 251
M10.2. LaplaceTransform ................................................. 251
M10.2.1. LaplaceTransformandtheInverseLaplaceTransform ......... 251
M10.2.2. MainPropertiesoftheLaplaceTransform. Inversion Formulasfor
SomeFunctions ........................................... 252
M10.2.3. Limit Theorems. Representation of Inverse Transforms as
Convergent Series ......................................... 255
M10.3. VariousFormsoftheFourierTransform ............................... 255
M10.3.1. FourierTransformandtheInverseFourierTransform .......... 255
M10.3.2. FourierCosineandSineTransforms ......................... 257
M10.4. MellinTransformandOtherTransforms ............................... 258
M10.4.1. MellinTransform andtheInversion Formula .................. 258
M10.4.2. Main Properties of the Mellin Transform. Relation Among the
Mellin,Laplace,andFourierTransforms ..................... 259
M10.4.3. SummaryTableofIntegralTransforms ....................... 259
Bibliography forChapterM10 .............................................. 260
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