Handbook of mathematics for engineers and scientists

1543 Pages·2006·9.468 MB·English

by  Andrei D. Polyanin, Alexander V. Manzhirov

HANDBOOK OF
MATHEMATICS FOR
ENGINEERS AND
SCIENTISTS
C5025_C000a.indd 1 10/16/06 2:53:21 PM
C5025_C000a.indd 2 10/16/06 2:53:21 PM
HANDBOOK OF
MATHEMATICS FOR
ENGINEERS AND
SCIENTISTS
Andrei D. Polyanin
Alexander V. Manzhirov
C5025_C000a.indd 3 10/16/06 2:53:21 PM
Chapman & Hall/CRC
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© 2007 by Taylor & Francis Group, LLC
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International Standard Book Number‑10: 1‑58488‑502‑5 (Hardcover)
International Standard Book Number‑13: 978‑1‑58488‑502‑3 (Hardcover)
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CONTENTS
Authors .................................................................. xxv
Preface ...................................................................xxvii
MainNotation ............................................................. xxix
PartI. Definitions,Formulas,Methods,andTheorems 1
1. ArithmeticandElementaryAlgebra ......................................... 3
1.1. RealNumbers ........................................................... 3
1.1.1. IntegerNumbers ................................................... 3
1.1.2. Real,Rational,andIrrationalNumbers ................................. 4
1.2. EqualitiesandInequalities. ArithmeticOperations. AbsoluteValue ................ 5
1.2.1. EqualitiesandInequalities ........................................... 5
1.2.2. AdditionandMultiplicationofNumbers ................................ 6
1.2.3. RatiosandProportions .............................................. 6
1.2.4. Percentage ....................................................... 7
1.2.5. AbsoluteValueofaNumber(ModulusofaNumber) ...................... 8
1.3. PowersandLogarithms ................................................... 8
1.3.1. PowersandRoots .................................................. 8
1.3.2. Logarithms ....................................................... 9
1.4. BinomialTheoremandRelatedFormulas ..................................... 10
1.4.1. Factorials. BinomialCoefficients. BinomialTheorem ..................... 10
1.4.2. RelatedFormulas .................................................. 10
1.5. ArithmeticandGeometricProgressions. FiniteSumsandProducts ................. 11
1.5.1. ArithmeticandGeometricProgressions ................................ 11
1.5.2. FiniteSeriesandProducts ........................................... 12
1.6. MeanValuesandInequalitiesofGeneralForm ................................. 13
1.6.1. ArithmeticMean,GeometricMean,andOtherMeanValues. Inequalitiesfor
MeanValues ...................................................... 13
1.6.2. InequalitiesofGeneralForm ......................................... 14
1.7. SomeMathematicalMethods .............................................. 15
1.7.1. ProofbyContradiction .............................................. 15
1.7.2. MathematicalInduction ............................................. 16
1.7.3. ProofbyCounterexample ........................................... 17
1.7.4. MethodofUndeterminedCoefficients .................................. 17
ReferencesforChapter1 ...................................................... 18
2. ElementaryFunctions ..................................................... 19
2.1. Power,Exponential,andLogarithmicFunctions ................................ 19
2.1.1. PowerFunction: y =xα ............................................ 19
2.1.2. ExponentialFunction: y =ax ........................................ 21
2.1.3. LogarithmicFunction: y =log x ..................................... 22
a
2.2. TrigonometricFunctions .................................................. 24
2.2.1. TrigonometricCircle. DefinitionofTrigonometricFunctions ............... 24
2.2.2. GraphsofTrigonometricFunctions .................................... 25
2.2.3. PropertiesofTrigonometricFunctions ................................. 27
v
vi CONTENTS
2.3. InverseTrigonometricFunctions ............................................ 30
2.3.1. Definitions. GraphsofInverseTrigonometricFunctions ................... 30
2.3.2. PropertiesofInverseTrigonometricFunctions ........................... 33
2.4. HyperbolicFunctions ..................................................... 34
2.4.1. Definitions. GraphsofHyperbolicFunctions ............................ 34
2.4.2. PropertiesofHyperbolicFunctions .................................... 36
2.5. InverseHyperbolicFunctions .............................................. 39
2.5.1. Definitions. GraphsofInverseHyperbolicFunctions ...................... 39
2.5.2. PropertiesofInverseHyperbolicFunctions .............................. 41
ReferencesforChapter2 ...................................................... 42
3. ElementaryGeometry .................................................... 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. SphericalTrigonometry ................................................... 70
3.3.1. SphericalGeometry ................................................ 70
3.3.2. SphericalTriangles ................................................ 71
ReferencesforChapter3 ...................................................... 75
4. AnalyticGeometry ....................................................... 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-OrderCurves ..................................................... 97
4.4.1. Circle ........................................................... 97
4.4.2. Ellipse .......................................................... 98
4.4.3. Hyperbola ........................................................ 101
4.4.4. Parabola ......................................................... 104
4.4.5. TransformationofSecond-OrderCurvestoCanonicalForm ................ 107
4.5. Coordinates,Vectors,Curves,andSurfacesinSpace ............................ 113
4.5.1. Vectors. CartesianCoordinateSystem ................................. 113
4.5.2. CoordinateSystems ................................................ 114
4.5.3. Vectors. ProductsofVectors ......................................... 120
4.5.4. CurvesandSurfacesinSpace ........................................ 123
CONTENTS vii
4.6. LineandPlaneinSpace ................................................... 124
4.6.1. PlaneinSpace .................................................... 124
4.6.2. LineinSpace ..................................................... 131
4.6.3. MutualArrangementofPoints,Lines,andPlanes ........................ 135
4.7. QuadricSurfaces(Quadrics) ............................................... 143
4.7.1. Quadrics(CanonicalEquations) ...................................... 143
4.7.2. Quadrics(GeneralTheory) .......................................... 148
ReferencesforChapter4 ...................................................... 153
5. Algebra ................................................................. 155
5.1. PolynomialsandAlgebraicEquations ........................................ 155
5.1.1. PolynomialsandTheirProperties ..................................... 155
5.1.2. LinearandQuadraticEquations ....................................... 157
5.1.3. CubicEquations ................................................... 158
5.1.4. Fourth-DegreeEquation ............................................. 159
5.1.5. AlgebraicEquationsofArbitraryDegreeandTheirProperties .............. 161
5.2. MatricesandDeterminants ................................................ 167
5.2.1. Matrices ......................................................... 167
5.2.2. Determinants ..................................................... 175
5.2.3. EquivalentMatrices. Eigenvalues ..................................... 180
5.3. LinearSpaces ........................................................... 187
5.3.1. ConceptofaLinearSpace. ItsBasisandDimension ...................... 187
5.3.2. SubspacesofLinearSpaces .......................................... 190
5.3.3. CoordinateTransformationsCorrespondingtoBasisTransformationsinaLinear
Space ........................................................... 191
5.4. EuclideanSpaces ........................................................ 192
5.4.1. RealEuclideanSpace ............................................... 192
5.4.2. ComplexEuclideanSpace(UnitarySpace) .............................. 195
5.4.3. BanachSpacesandHilbertSpaces .................................... 196
5.5. SystemsofLinearAlgebraicEquations ....................................... 197
5.5.1. ConsistencyConditionforaLinearSystem ............................. 197
5.5.2. FindingSolutionsofaSystemofLinearEquations ....................... 198
5.6. LinearOperators ........................................................ 204
5.6.1. NotionofaLinearOperator. ItsProperties .............................. 204
5.6.2. LinearOperatorsinMatrixForm ...................................... 208
5.6.3. EigenvectorsandEigenvaluesofLinearOperators ........................ 209
5.7. BilinearandQuadraticForms .............................................. 213
5.7.1. LinearandSesquilinearForms ....................................... 213
5.7.2. BilinearForms .................................................... 214
5.7.3. QuadraticForms ................................................... 216
5.7.4. BilinearandQuadraticFormsinEuclideanSpace ........................ 219
5.7.5. Second-OrderHypersurfaces ......................................... 220
5.8. SomeFactsfromGroupTheory ............................................ 225
5.8.1. GroupsandTheirBasicProperties .................................... 225
5.8.2. TransformationGroups ............................................. 228
5.8.3. GroupRepresentations .............................................. 230
ReferencesforChapter5 ...................................................... 233
viii CONTENTS
6. LimitsandDerivatives .................................................... 235
6.1. BasicConceptsofMathematicalAnalysis .................................... 235
6.1.1. NumberSets. FunctionsofRealVariable ............................... 235
6.1.2. LimitofaSequence ................................................ 237
6.1.3. LimitofaFunction. Asymptotes ...................................... 240
6.1.4. InfinitelySmallandInfinitelyLargeFunctions ........................... 242
6.1.5. ContinuousFunctions. DiscontinuitiesoftheFirstandtheSecondKind ....... 243
6.1.6. ConvexandConcaveFunctions ....................................... 245
6.1.7. FunctionsofBoundedVariation ...................................... 246
6.1.8. ConvergenceofFunctions ........................................... 249
6.2. DifferentialCalculusforFunctionsofaSingleVariable .......................... 250
6.2.1. DerivativeandDifferential,TheirGeometricalandPhysicalMeaning ......... 250
6.2.2. TableofDerivativesandDifferentiationRules ........................... 252
6.2.3. TheoremsaboutDifferentiableFunctions. L’HospitalRule ................. 254
6.2.4. Higher-OrderDerivativesandDifferentials. Taylor’sFormula ............... 255
6.2.5. ExtremalPoints. PointsofInflection ................................... 257
6.2.6. QualitativeAnalysisofFunctionsandConstructionofGraphs .............. 259
6.2.7. ApproximateSolutionofEquations(Root-FindingAlgorithmsforContinuous
Functions) ....................................................... 260
6.3. FunctionsofSeveralVariables. PartialDerivatives .............................. 263
6.3.1. PointSets. Functions. LimitsandContinuity ............................ 263
6.3.2. DifferentiationofFunctionsofSeveralVariables ......................... 264
6.3.3. DirectionalDerivative. Gradient. GeometricalApplications ................ 267
6.3.4. ExtremalPointsofFunctionsofSeveralVariables ........................ 269
6.3.5. DifferentialOperatorsoftheFieldTheory .............................. 272
ReferencesforChapter6 ...................................................... 272
7. Integrals ................................................................ 273
7.1. IndefiniteIntegral ........................................................ 273
7.1.1. Antiderivative. IndefiniteIntegralandItsProperties ....................... 273
7.1.2. TableofBasicIntegrals. PropertiesoftheIndefiniteIntegral. Integration
Examples ........................................................ 274
7.1.3. IntegrationofRationalFunctions ..................................... 276
7.1.4. IntegrationofIrrationalFunctions ..................................... 279
7.1.5. IntegrationofExponentialandTrigonometricFunctions ................... 281
7.1.6. IntegrationofPolynomialsMultipliedbyElementaryFunctions ............. 283
7.2. DefiniteIntegral ......................................................... 286
7.2.1. BasicDefinitions. ClassesofIntegrableFunctions. GeometricalMeaningofthe
DefiniteIntegral ................................................... 286
7.2.2. PropertiesofDefiniteIntegralsandUsefulFormulas ...................... 287
7.2.3. GeneralReductionFormulasfortheEvaluationofIntegrals ................ 289
7.2.4. GeneralAsymptoticFormulasfortheCalculationofIntegrals ............... 290
7.2.5. MeanValueTheorems. PropertiesofIntegralsinTermsofInequalities.
ArithmeticMeanandGeometricMeanofFunctions ...................... 295
7.2.6. GeometricandPhysicalApplicationsoftheDefiniteIntegral ............... 299
7.2.7. ImproperIntegralswithInfiniteIntegrationLimit ......................... 301
7.2.8. GeneralReductionFormulasfortheCalculationofImproperIntegrals ........ 304
7.2.9. GeneralAsymptoticFormulasfortheCalculationofImproperIntegrals ....... 307
7.2.10. ImproperIntegralsofUnboundedFunctions ............................ 308
7.2.11. Cauchy-TypeSingularIntegrals ...................................... 310
CONTENTS ix
7.2.12. StieltjesIntegral .................................................. 312
7.2.13. SquareIntegrableFunctions ........................................ 314
7.2.14. Approximate(Numerical)MethodsforComputationofDefiniteIntegrals .... 315
7.3. DoubleandTripleIntegrals ................................................ 317
7.3.1. DefinitionandPropertiesoftheDoubleIntegral .......................... 317
7.3.2. ComputationoftheDoubleIntegral ................................... 319
7.3.3. GeometricandPhysicalApplicationsoftheDoubleIntegral ................ 323
7.3.4. DefinitionandPropertiesoftheTripleIntegral ........................... 324
7.3.5. ComputationoftheTripleIntegral. SomeApplications. IteratedIntegralsand
AsymptoticFormulas ............................................... 325
7.4. LineandSurfaceIntegrals ................................................. 329
7.4.1. LineIntegraloftheFirstKind ........................................ 329
7.4.2. LineIntegraloftheSecondKind ...................................... 330
7.4.3. SurfaceIntegraloftheFirstKind ..................................... 332
7.4.4. SurfaceIntegraloftheSecondKind ................................... 333
7.4.5. IntegralFormulasofVectorCalculus .................................. 334
ReferencesforChapter7 ...................................................... 335
8. Series .................................................................. 337
8.1. NumericalSeriesandInfiniteProducts ....................................... 337
8.1.1. ConvergentNumericalSeriesandTheirProperties. Cauchy’sCriterion ....... 337
8.1.2. ConvergenceCriteriaforSerieswithPositive(Nonnegative)Terms .......... 338
8.1.3. ConvergenceCriteriaforArbitraryNumericalSeries. AbsoluteandConditional
Convergence ...................................................... 341
8.1.4. MultiplicationofSeries. SomeInequalities ............................. 343
8.1.5. SummationMethods. ConvergenceAcceleration ......................... 344
8.1.6. InfiniteProducts ................................................... 346
8.2. FunctionalSeries ........................................................ 348
8.2.1. PointwiseandUniformConvergenceofFunctionalSeries .................. 348
8.2.2. BasicCriteriaofUniformConvergence. PropertiesofUniformlyConvergent
Series ........................................................... 349
8.3. PowerSeries ............................................................ 350
8.3.1. RadiusofConvergenceofPowerSeries. PropertiesofPowerSeries .......... 350
8.3.2. TaylorandMaclaurinPowerSeries .................................... 352
8.3.3. OperationswithPowerSeries. SummationFormulasforPowerSeries ........ 354
8.4. FourierSeries ........................................................... 357
8.4.1. Representationof2π-PeriodicFunctionsbyFourierSeries. MainResults ..... 357
8.4.2. FourierExpansionsofPeriodic,Nonperiodic,Odd,andEvenFunctions ....... 359
8.4.3. CriteriaofUniformandMean-SquareConvergenceofFourierSeries ......... 361
8.4.4. SummationFormulasforTrigonometricSeries ........................... 362
8.5. AsymptoticSeries ....................................................... 363
8.5.1. AsymptoticSeriesofPoincare´Type. FormulasfortheCoefficients ........... 363
8.5.2. OperationswithAsymptoticSeries .................................... 364
ReferencesforChapter8 ...................................................... 366
9. DifferentialGeometry ..................................................... 367
9.1. TheoryofCurves ........................................................ 367
9.1.1. PlaneCurves ..................................................... 367
9.1.2. SpaceCurves ..................................................... 379