Mathematics for Physicists and Engineers

Transcription

1 Mathematics for Physicists and Engineers

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

3 Prof. Dr. Klaus Weltner University of Frankfurt Institute for Didactic of Physics Frankfurt, Germany Sebastian John Jahnsdorf, Germany Dr. Peter Schuster Sulzbach, Germany Wolfgang J. Weber University of Frankfurt Computing Center Frankfurt, Germany Prof. Dr. Jean Grosjean School of Engineering at the University of Bath Great Britain Originally published in German language under the title Mathematik für Physiker und Ingenieure, Bände 1 und 2 by K. Weltner in collaboration with 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, S. John and K. Weltner. Cartoons in the study guide by Martin Weltner. Additional Material is available under extras.springer.com Password: ISBN ISBN (ebook) DOI / Springer Heidelberg New York Dordrecht London Library of Congress Control Number: Springer-Verlag Berlin Heidelberg 2009, 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. 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. 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. Dipl. Phys. Sebastian John is postgraduate at the University Cambridge (England) v

5 Preface and Introduction The Combination of Textbook and Study Guide This textbook is a combination of a printed book and a study guide presented online. The volume of the study guide is nearly twice that of the printed textbook. Both parts are of equal importance. 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 vii

6 viii Main Authors of the International Version 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. In this second international edition printing errors have been corrected and explanations have been improved. Furthermore a new chapter Fourier Integrals and Fourier Transforms has been included. 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, 2013 K. Weltner J. Grosjean W. J. Weber S. John P. Schuster How to Access the Study Guide The study guide is available for all readers of the books by the internet. To access the study guide click extras.springer.com. Then you are asked the ISBN number. It is Now download the entire content or single files. Then extract the download. The study guide is presented in two versions. In a first version the study guide is presented on the screen of your PC. You will work online. In this case you follow the instructions given on the screen, at first studying sections of the textbook off the PC. After this autonomous study period you are to answer questions and to solve problems presented online by the PC. A second version is given as pdf files for students who prefer to work with printed texts. In this case the chosen chapter has to be printed. To use the study guide all necessary explanations are given by the text.

7 Contents Preface and Introduction... vii 1 Vector Algebra I: Scalars and Vectors Scalars and Vectors Addition of Vectors Sum of Two Vectors: Geometrical Addition Subtraction of Vectors Components and Projection of a Vector Component Representation in Coordinate Systems Position Vector Unit Vectors 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 Magnitude of a Vector Vector Algebra II: Scalar and Vector Products Scalar Product Application: Equation of a Line and a Plane Special Cases Commutative and Distributive Laws Scalar Product in Terms of the Components of the Vectors Vector Product Torque Torque as a Vector Definition of the Vector Product Special Cases Anti-Commutative Law for Vector Products Components of the Vector Product ix

8 x Contents 3 Functions The Mathematical Concept of Functions and its Meaning in Physics and Engineering Introduction The Concept of a Function Graphical Representation of Functions Coordinate System, Position Vector The Linear Function: The Straight Line Graph Plotting Quadratic Equations Parametric Changes of Functions and Their Graphs Inverse Functions Trigonometric or Circular Functions Unit Circle Sine Function Cosine Function Relationships Between the Sine and Cosine Functions Tangent and Cotangent Addition Formulae Inverse Trigonometric Functions Function of a Function (Composition) Exponential, Logarithmic and Hyperbolic Functions Powers, Exponential Function Powers Laws of Indices or Exponents Binomial Theorem Exponential Function Logarithm, Logarithmic Function Logarithm Operations with Logarithms Logarithmic Functions Hyperbolic Functions and Inverse Hyperbolic Functions Hyperbolic Functions Inverse Hyperbolic Functions Differential Calculus Sequences and Limits The Concept of Sequence Limit of a Sequence Limit of a Function Examples for the Practical Determination of Limits Continuity... 93

9 Contents xi 5.3 Series Geometric Series Differentiation of a Function Gradient or Slope of a Line Gradient of an Arbitrary Curve Derivative of a Function Physical Application: Velocity The Differential Calculating Differential Coefficients Derivatives of Power Functions; Constant Factors Rules for Differentiation Differentiation of Fundamental Functions Higher Derivatives Extreme Values and Points of Inflexion; Curve Sketching Maximum and Minimum Values of a Function Further Remarks on Points of Inflexion (Contraflexure) Curve Sketching Applications of Differential Calculus Extreme Values Increments Curvature Determination of Limits by Differentiation: L Hôpital s Rule Further Methods for Calculating Differential Coefficients Implicit Functions and their Derivatives Logarithmic Differentiation Parametric Functions and their Derivatives Parametric Form of an Equation Derivatives of Parametric Functions Integral Calculus The Primitive Function Fundamental Problem of Integral Calculus The Area Problem: The Definite Integral Fundamental Theorem of the Differential and Integral Calculus The Definite Integral Calculation of Definite Integrals from Indefinite Integrals Examples of Definite Integrals Methods of Integration Principle of Verification Standard Integrals...161

10 xii Contents Constant Factor and the Sum of Functions Integration by Parts: Product of Two Functions Integration by Substitution Substitution in Particular Cases Integration by Partial Fractions Rules for Solving Definite Integrals Mean Value Theorem Improper Integrals Line Integrals Applications of Integration Areas Areas for Parametric Functions Areas in Polar Coordinates Areas of Closed Curves Lengths of Curves Lengths of Curves in Polar Coordinates Surface Area and Volume of a Solid of Revolution Applications to Mechanics Basic Concepts of Mechanics Center of Mass and Centroid The Theorems of Pappus Moments of Inertia; Second Moment of Area Taylor Series and Power Series Introduction Expansion of a Function in a Power Series Interval of Convergence of Power Series Approximate Values of Functions Expansion of a Function f (x) at an Arbitrary Position Applications of Series Polynomials as Approximations Integration of Functions when Expressed as Power Series Expansion in a Series by Integrating Complex Numbers Definition and Properties of Complex Numbers Imaginary Numbers Complex Numbers Fields of Application Operations with Complex Numbers Graphical Representation of Complex Numbers Gauss Complex Number Plane: Argand Diagram Polar Form of a Complex Number...253

11 Contents xiii 9.3 Exponential Form of Complex Numbers Euler s Formula Exponential Form of the Sine and Cosine Functions Complex Numbers as Powers Multiplication and Division in Exponential Form Raising to a Power, Exponential Form Periodicity of re j Transformation of a Complex Number From One Form into Another Operations with Complex Numbers Expressed in Polar Form Multiplication and Division Raising to a Power Roots of a Complex Number Differential Equations Concept and Classification of Differential Equations Preliminary Remarks General Solution of First- and Second-Order DEs with Constant Coefficients Homogeneous Linear DE Non-Homogeneous Linear DE Boundary Value Problems First-Order DEs Second-Order DEs Some Applications of DEs Radioactive Decay The Harmonic Oscillator General Linear First-Order DEs Solution by Variation of the Constant A Straightforward Method Involving the Integrating Factor Some Remarks on General First-Order DEs Bernoulli s Equations Separation of Variables Exact Equations The Integrating Factor General Case Simultaneous DEs Higher-Order DEs Interpreted as Systems of First-Order Simultaneous DEs Some Advice on Intractable DEs Laplace Transforms Introduction The Laplace Transform Definition Laplace Transform of Standard Functions...324

12 xiv Contents 11.4 Solution of Linear DEs with Constant Coefficients Solution of Simultaneous DEs with Constant Coefficients Functions of Several Variables; Partial Differentiation; and Total Differentiation Introduction Functions of Several Variables Representing the Surface by Establishing a Table of Z-Values Representing the Surface by Establishing Intersecting Curves Obtaining a Functional Expression for a Given Surface Partial Differentiation Higher Partial Derivatives Total Differential Total Differential of Functions Application: Small Tolerances Gradient Total Derivative Explicit Functions Implicit Functions Maxima and Minima of Functions of Two or More Variables Applications: Wave Function and Wave Equation Wave Function Wave Equation 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 Coordinate Systems Polar Coordinates Cylindrical Coordinates Spherical Coordinates Application: Moments of Inertia of a Solid Transformation of Coordinates; Matrices Introduction Parallel Shift of Coordinates: Translation Rotation Rotation in a Plane Successive Rotations Rotations in Three-Dimensional Space...413

13 Contents xv 14.4 Matrix Algebra Addition and Subtraction of Matrices Multiplication of a Matrix by a Scalar Product of a Matrix and a Vector Multiplication of Two Matrices Rotations Expressed in Matrix Form Rotation in Two-Dimensional Space Special Rotation in Three-Dimensional Space Special Matrices Inverse Matrix Sets of Linear Equations; Determinants Introduction Sets of Linear Equations Gaussian Elimination: Successive Elimination of Variables Gauss Jordan Elimination Matrix Notation of Sets of Equations and Determination of the Inverse Matrix Existence of Solutions Determinants Preliminary Remarks on Determinants Definition and Properties of an n-row Determinant Rank of a Determinant and Rank of a Matrix Applications of Determinants Eigenvalues and Eigenvectors of Real Matrices Two Case Studies: Eigenvalues of 2 2 Matrices General Method for Finding Eigenvalues Worked Example: Eigenvalues of a 3 3 Matrix Important Facts on Eigenvalues and Eigenvectors Vector Analysis: Surface Integrals, Divergence, Curl and Potential Flow of a Vector Field Through a Surface Element Surface Integral Special Cases of Surface Integrals Flow of a Homogeneous Vector Field Through a Cuboid Flow of a Spherically Symmetrical Field Through a Sphere Application: The Electrical Field of a Point Charge General Case of Computing Surface Integrals Divergence of a Vector Field Gauss s Theorem Curl of a Vector Field Stokes Theorem...486

14 xvi Contents 17.9 Potential of a Vector Field Short Reference on Vector Derivatives Fourier Series; Harmonic Analysis Expansion of a Periodic Function into a Fourier Series Evaluation of the Coefficients Odd and Even Functions Examples of Fourier Series Expansion of Functions of Period 2L Fourier Spectrum Fourier Integrals and Fourier Transforms Transition from Fourier Series to Fourier Integral Fourier Transforms Fourier Cosine Transform Fourier Sine Transform, General Fourier Transform Complex Representation of the Fourier Transform Shift Theorem Discrete Fourier Transform, Sampling Theorems Fourier Transform of the Gaussian Function 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 Permutations and Combinations 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 of the Binomial Distribution Properties of the Normal Distribution Derivation of the Binomial Distribution...544

15 Contents xvii 22 Theory of Errors Purpose of the Theory of Errors Mean Value and Variance Mean Value Variance and Standard Deviation Mean Value and Variance in a Random Sample and Parent Population Mean Value and Variance of Continuous Distributions Error in Mean Value Normal Distribution: Distribution of Random Errors Law of Error Propagation Weighted Average Curve Fitting: Method of Least Squares, Regression Line Correlation and Correlation Coefficient Answers Index...595