Mathematics for Physicists and Engineers

Transcription

1 Mathematics for Physicists and Engineers

2 Klaus Weltner Wolfgang J. Weber Jean Grosjean Peter Schuster Mathematics for Physicists and Engineers Fundamentals and Interactive Study Guide 123

3 Prof. Dr. Klaus Weltner University of Frankfurt Institute for Didactic of Physics Max-von-Laue-Straße Frankfurt/Main Germany Weltner@em.uni-frankfurt.de Wolfgang J. Weber University of Frankfurt Computing Center Grüneburgplatz Frankfurt Germany weber@rz.uni-frankfurt.de Dr. Peter Schuster Am Holzweg Sulzbach Germany Prof. Dr. Jean Grosjean School of Engineering at the University of Bath England This title was originally published by Stanley Thornes (Publisher) Ltd, 1986, entitled Mathematics for Engineers and Scientists by K. Weltner, J. Grosjean, F. Schuster and W.J. Weber. Cartoons in the study guide by Martin Weltner. ISBN e-isbn DOI / Springer Dordrecht Heidelberg London New York Library of Congress Control Number: Springer-Verlag Berlin Heidelberg 2009 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher and the authors accept no legal responsibility for any damage caused by improper use of the instructions and programs contained in this book and the CD. Although the software has been tested with extreme care, errors in the software cannot be excluded. Typesetting and Production: le-tex publishing services GmbH, Leipzig, Germany Cover design: estudio Calamar S.L., Spain/Germany Printed on acid-free paper Springer is part of Springer Science+Business Media (

4 Main Authors of the International Version Prof. Dr. Klaus Weltner has studied physics at the Technical University Hannover (Germany) and the University of Bristol (England). He graduated in plasma physics and was professor of physics and didactic of physics at the universities Osnabrück, Berlin, Frankfurt and visiting professor of physics at the Federal University of Bahia (Brazil). Prof. Dr. Jean Grosjean was Head of Applied Mechanics at the School of Engineering at the University of Bath (England). Wolfgang J. Weber has studied mathematics at the universities of Frankfurt (Germany), Oxford (England) and Michigan State (USA). He is currently responsible for the training of computer specialists at the computing center at the University of Frankfurt. Dr.-Ing. Peter Schuster was lecturer at the School of Engineering at the University of Bath (England). Different appointments in the chemical industry. v

5 Preface Mathematics is an essential tool for physicists and engineers which students must use from the very beginning of their studies. This combination of textbook and study guide aims to develop as rapidly as possible the students ability to understand and to use those parts of mathematics which they will most frequently encounter. Thus functions, vectors, calculus, differential equations and functions of several variables are presented in a very accessible way. Further chapters in the book provide the basic knowledge on various important topics in applied mathematics. Based on their extensive experience as lecturers, each of the authors has acquired a close awareness of the needs of first- and second-years students. One of their aims has been to help users to tackle successfully the difficulties with mathematics which are commonly met. A special feature which extends the supportive value of the main textbook is the accompanying study guide. This study guide aims to satisfy two objectives simultaneously: it enables students to make more effective use of the main textbook, and it offers advice and training on the improvement of techniques on the study of textbooks generally. The study guide divides the whole learning task into small units which the student is very likely to master successfully. Thus he or she is asked to read and study a limited section of the textbook and to return to the study guide afterwards. Learning results are controlled, monitored and deepened by graded questions, exercises, repetitions and finally by problems and applications of the content studied. Since the degree of difficulties is slowly rising the students gain confidence immediately and experience their own progress in mathematical competence thus fostering motivation. In case of learning difficulties he or she is given additional explanations and in case of individual needs supplementary exercises and applications. So the sequence of the studies is individualised according to the individual performance and needs and can be regarded as a full tutorial course. The work was originally published in Germany under the title Mathematik für Physiker (Mathematics for physicists). It has proved its worth in years of actual use. This new international version has been modified and extended to meet the needs of students in physics and engineering. vii

6 viii Preface The CD offers two versions. In a first version the frames of the study guide are presented on a PC screen. In this case the user follows the instructions given on the screen, at first studying sections of the textbook off the PC. After this autonomous study he is to answer questions and to solve problems presented by the PC. A second version is given as pdf files for students preferring to work with a print version. Both the textbook and the study guide have resulted from teamwork. The authors of the original textbook and study guides were Prof. Dr. Weltner, Prof. Dr. P.-B. Heinrich, Prof. Dr. H. Wiesner, P. Engelhard and Prof. Dr. H. Schmidt. The translation and the adaption was undertaken by the undersigned. Frankfurt, August 2009 K. Weltner J. Grosjean P. Schuster W. J. Weber

7 Acknowledgement Originally published in the Federal Republic of Germany under the title Mathematik für Physiker by the authors K. Weltner, H. Wiesner, P.-B. Heinrich, P. Engelhardt and H. Schmidt. The work has been translated by J. Grosjean and P. Schuster and adapted to the needs of engineering and science students in English speaking countries by J. Grosjean, P. Schuster, W.J. Weber and K. Weltner. ix

8 Contents Preface... vii 1 Vector Algebra I: Scalars and Vectors ScalarsandVectors Addition of Vectors Sum of Two Vectors: Geometrical Addition SubtractionofVectors Components and Projection of a Vector Component Representation in Coordinate Systems Position Vector UnitVectors Component Representation of a Vector Representation of the Sum of Two Vectors in Terms of Their Components Subtraction of Vectors in Terms of their Components Multiplication of a Vector by a Scalar MagnitudeofaVector Vector Algebra II: Scalar and Vector Products Scalar Product Application:EquationofaLineandaPlane SpecialCases CommutativeandDistributiveLaws Scalar Product in Terms of the Components of the Vectors Vector Product Torque TorqueasaVector Definition of the Vector Product SpecialCases Anti-Commutative Law for Vector Products Components of the Vector Product xi

9 xii Contents 3 Functions The Mathematical Concept of Functions anditsmeaninginphysicsandengineering Introduction TheConceptofaFunction GraphicalRepresentationofFunctions Coordinate System, Position Vector TheLinearFunction:TheStraightLine Graph Plotting QuadraticEquations ParametricChangesofFunctionsandTheirGraphs InverseFunctions Trigonometric or Circular Functions UnitCircle SineFunction CosineFunction Relationships Between the Sine and Cosine Functions TangentandCotangent Addition Formulae Inverse Trigonometric Functions FunctionofaFunction(Composition) Exponential, Logarithmic and Hyperbolic Functions Powers, Exponential Function Powers Laws of Indices or Exponents BinomialTheorem Exponential Function Logarithm,LogarithmicFunction Logarithm OperationswithLogarithms LogarithmicFunctions HyperbolicFunctionsandInverseHyperbolicFunctions HyperbolicFunctions InverseHyperbolicFunctions Differential Calculus SequencesandLimits TheConceptofSequence LimitofaSequence LimitofaFunction ExamplesforthePracticalDeterminationofLimits Continuity... 91

10 Contents xiii 5.3 Series GeometricSeries DifferentiationofaFunction GradientorSlopeofaLine GradientofanArbitraryCurve DerivativeofaFunction PhysicalApplication:Velocity TheDifferential CalculatingDifferentialCoefficients DerivativesofPowerFunctions;ConstantFactors RulesforDifferentiation Differentiation of Fundamental Functions HigherDerivatives ExtremeValuesandPointsofInflexion;CurveSketching MaximumandMinimumValuesofaFunction Further Remarks on Points of Inflexion (Contraflexure) CurveSketching ApplicationsofDifferentialCalculus ExtremeValues Increments Curvature Determination of Limits by Differentiation: L Hôpital srule Further Methods for Calculating Differential Coefficients ImplicitFunctionsandtheirDerivatives LogarithmicDifferentiation ParametricFunctionsandtheirDerivatives ParametricFormofanEquation DerivativesofParametricFunctions Integral Calculus ThePrimitiveFunction Fundamental Problem of Integral Calculus TheAreaProblem:TheDefiniteIntegral Fundamental Theorem ofthedifferentialandintegralcalculus TheDefiniteIntegral Calculation of Definite Integrals from Indefinite Integrals ExamplesofDefiniteIntegrals Methods of Integration PrincipleofVerification StandardIntegrals...159

11 xiv Contents ConstantFactorandtheSumofFunctions Integration by Parts: Product of Two Functions Integration by Substitution Substitution in Particular Cases IntegrationbyPartialFractions RulesforSolvingDefiniteIntegrals MeanValueTheorem ImproperIntegrals LineIntegrals Applications of Integration Areas AreasforParametricFunctions AreasinPolarCoordinates AreasofClosedCurves LengthsofCurves LengthsofCurvesinPolarCoordinates SurfaceAreaandVolumeofaSolidofRevolution ApplicationstoMechanics BasicConceptsofMechanics CenterofMassandCentroid The Theorems of Pappus MomentsofInertia;SecondMomentofArea Taylor Series and Power Series Introduction ExpansionofaFunctioninaPowerSeries IntervalofConvergenceofPowerSeries ApproximateValuesofFunctions Expansion of a Function f (x) at an Arbitrary Position ApplicationsofSeries Polynomials as Approximations Integration of Functions when Expressed as Power Series ExpansioninaSeriesbyIntegrating Complex Numbers Definition and Properties of Complex Numbers ImaginaryNumbers ComplexNumbers FieldsofApplication OperationswithComplexNumbers GraphicalRepresentationofComplexNumbers GaussComplexNumberPlane:ArgandDiagram PolarFormofaComplexNumber...251

12 Contents xv 9.3 Exponential Form of Complex Numbers Euler sformula Exponential Form of the Sine and Cosine Functions ComplexNumbersasPowers Multiplication and Division in Exponential Form Raising to a Power, Exponential Form Periodicity of re j Transformation of a Complex Number From One Form intoanother OperationswithComplexNumbersExpressedinPolarForm Multiplication and Division RaisingtoaPower RootsofaComplexNumber Differential Equations ConceptandClassificationofDifferentialEquations PreliminaryRemarks General Solution of First- and Second-Order DEs withconstantcoefficients HomogeneousLinearDE Non-HomogeneousLinearDE Boundary Value Problems First-OrderDEs Second-Order DEs SomeApplicationsofDEs Radioactive Decay The Harmonic Oscillator GeneralLinearFirst-OrderDEs SolutionbyVariationoftheConstant A Straightforward Method Involving the Integrating Factor SomeRemarksonGeneralFirst-OrderDEs Bernoulli s Equations SeparationofVariables ExactEquations TheIntegratingFactor GeneralCase SimultaneousDEs Higher-Order DEs Interpreted as Systems offirst-ordersimultaneousdes SomeAdviceonIntractableDEs Laplace Transforms Introduction The Laplace Transform Definition LaplaceTransformofStandardFunctions...322

13 xvi Contents 11.4 SolutionofLinearDEswithConstantCoefficients SolutionofSimultaneousDEswithConstantCoefficients Functions of Several Variables; Partial Differentiation; and Total Differentiation Introduction FunctionsofSeveralVariables Representing the Surface by Establishing a Table of Z-Values Representing the Surface byestablishingintersectingcurves ObtainingaFunctionalExpressionforaGivenSurface PartialDifferentiation HigherPartialDerivatives TotalDifferential TotalDifferentialofFunctions Application:SmallTolerances Gradient TotalDerivative ExplicitFunctions ImplicitFunctions Maxima and Minima of Functions of Two or More Variables Applications:WaveFunctionandWaveEquation WaveFunction WaveEquation Multiple Integrals; Coordinate Systems Multiple Integrals Multiple Integrals with Constant Limits Decomposition of a Multiple Integral into a Product of Integrals Multiple Integrals with Variable Limits CoordinateSystems PolarCoordinates CylindricalCoordinates SphericalCoordinates Application:MomentsofInertiaofaSolid Transformation of Coordinates; Matrices Introduction ParallelShiftofCoordinates:Translation Rotation RotationinaPlane SuccessiveRotations RotationsinThree-DimensionalSpace...411

14 Contents xvii 14.4 MatrixAlgebra Addition and Subtraction of Matrices Multiplication of a Matrix by a Scalar Product of a Matrix and a Vector Multiplication of Two Matrices RotationsExpressedinMatrixForm RotationinTwo-DimensionalSpace SpecialRotationinThree-DimensionalSpace SpecialMatrices InverseMatrix Sets of Linear Equations; Determinants Introduction SetsofLinearEquations Gaussian Elimination: Successive Elimination of Variables Gauss JordanElimination Matrix Notation of Sets of Equations and Determination oftheinversematrix ExistenceofSolutions Determinants PreliminaryRemarksonDeterminants Definition and Properties of an n-row Determinant RankofaDeterminantandRankofaMatrix ApplicationsofDeterminants Eigenvalues and Eigenvectors of Real Matrices Two Case Studies: Eigenvalues of 2 2 Matrices GeneralMethodforFindingEigenvalues Worked Example: Eigenvalues of a 3 3 Matrix ImportantFactsonEigenvaluesandEigenvectors Vector Analysis: Surface Integrals, Divergence, Curl and Potential Flow of a Vector Field Through a Surface Element SurfaceIntegral SpecialCasesofSurfaceIntegrals Flow of a Homogeneous Vector Field Through a Cuboid Flow of a Spherically Symmetrical Field Through asphere Application:TheElectricalFieldofaPointCharge GeneralCaseofComputingSurfaceIntegrals DivergenceofaVectorField Gauss stheorem CurlofaVectorField Stokes Theorem...484

15 xviii Contents 17.9 PotentialofaVectorField ShortReferenceonVectorDerivatives Fourier Series; Harmonic Analysis ExpansionofaPeriodicFunctionintoaFourierSeries EvaluationoftheCoefficients OddandEvenFunctions ExamplesofFourierSeries Expansion of Functions of Period 2L FourierSpectrum Probability Calculus Introduction Concept of Probability Random Experiment, Outcome Space and Events The Classical Definition of Probability The Statistical Definition of Probability General Properties of Probabilities Probability of Statistically Independent Events. Compound Probability PermutationsandCombinations Permutations Combinations Probability Distributions Discrete and Continuous Probability Distributions Discrete Probability Distributions Continuous Probability Distributions Mean Values of Discrete and Continuous Variables The Normal Distribution as the Limiting Value ofthebinomialdistribution PropertiesoftheNormalDistribution DerivationoftheBinomialDistribution Theory of Errors PurposeoftheTheoryofErrors MeanValueandVariance MeanValue VarianceandStandardDeviation Mean Value and Variance in a Random Sample and Parent Population Mean Value and Variance of Continuous Distributions ErrorinMeanValue Normal Distribution: Distribution of Random Errors LawofErrorPropagation...546

16 Contents xix 21.7 WeightedAverage Curve Fitting: Method of Least Squares, Regression Line CorrelationandCorrelationCoefficient Answers Index...581