Holdings: Mathematical methods

Mathematical methods: for students of physics and related fields

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Bibliographic Details
Main Author:	Hassani, Sadri (Author)
Format:	Book
Language:	English
Published:	New York [u.a.] Springer 2009
Edition:	2. ed.
Subjects:	Mathematische Physik Mathematical physics > Problems, exercises, etc Mathematical physics > Study and teaching Topologie Kategorie > Mathematik Konferenzschrift > 1978 > Berlin West
Online Access:	Inhaltsverzeichnis
Physical Description:	XXIII, 829 S. Ill., graph. Darst.
ISBN:	9780387095035

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adam_text	Contents Preface to Second Edition vii Preface ix Note to the Reader xiii I Coordinates and Calculus 1 1 Coordinate Systems and Vectors 3 1.1 Vectors in a Plane and in Space .................. 3 1.1.1 Dot Product ........................ 5 1.1.2 Vector or Cross Product .................. 7 1.2 Coordinate Systems ........................ 11 1.3 Vectors in Different Coordinate Systems ............. 16 1.3.1 Fields and Potentials .................... 21 1.3.2 Cross Product ....................... 28 1.4 Relations Among Unit Vectors .................. 31 1.5 Problems .............................. 37 2 Differentiation 43 2.1 The Derivative ........................... 44 2.2 Partial Derivatives ......................... 47 2.2.1 Definition, Notation, and Basic Properties ........ 47 2.2.2 Differentials ......................... 53 2.2.3 Chain Rule ......................... 55 2.2.4 Homogeneous Functions .................. 57 2.3 Elements of Length, Area, and Volume .............. 59 2.3.1 Elements in a Cartesian Coordinate System ....... 60 2.3.2 Elements in a Spherical Coordinate System ....... 62 2.3.3 Elements in a Cylindrical Coordinate System ...... 65 2.4 Problems .............................. 68 XVI CONTENTS Integration: Formalism 77 3.1 ƒ Means /um ......................... 77 3.2 Properties of Integral ........................ 81 3.2.1 Change of Dummy Variable ................ 82 3.2.2 Linearity .......................... 82 3.2.3 Interchange of Limits ................... 82 3.2.4 Partition of Range of Integration ............. 82 3.2.5 Transformation of Integration Variable .......... 83 3.2.6 Small Region of Integration ................ 83 3.2.7 Integral and Absolute Value ................ 84 3.2.8 Symmetric Range of Integration ............. 84 3.2.9 Differentiating an Integral ................. 85 3.2.10 Fundamental Theorem of Calculus ............ 87 3.3 Guidelines for Calculating Integrals ................ 91 3.3.1 Reduction to Single Integrals ............... 92 3.3.2 Components of Integrals of Vector Functions ...... 95 3.4 Problems .............................. 98 Integration: Applications 101 4.1 Single Integrals ........................... 101 4.1.1 An Example from Mechanics ............... 101 4.1.2 Examples from Electrostatics and Gravity ........ 104 4.1.3 Examples from Magnetostatics .............. 109 4.2 Applications: Double Integrals .................. 115 4.2.1 Cartesian Coordinates ................... 115 4.2.2 Cylindrical Coordinates .................. 118 4.2.3 Spherical Coordinates ................... 120 4.3 Applications: Triple Integrals ................... 122 4.4 Problems .............................. 128 Dirac Delta Function 139 5.1 One-Variable Case ......................... 139 5.1.1 Linear Densities of Points ................. 143 5.1.2 Properties of the Delta Function ............. 145 5.1.3 The Step Function ..................... 152 5.2 Two-Variable Case ......................... 154 5.3 Three-Variable Case ........................ 159 5.4 Problems .............................. 166 II Algebra of Vectors 171 6 Planar and Spatial Vectors 173 6.1 Vectors in a Plane Revisited ....................174 6.1.1 Transformation of Components ..............176 6.1.2 Inner Product ........................182 CONTENTS____________________________________________________________xvii 6.1.3 Orthogonal Transformation................190 6.2 Vectors in Space..........................192 6.2.1 Transformation of Vectors................. 194 6.2.2 Inner Product........................ 198 6.3 Determinant .............................202 6.4 The Jacobian ............................207 6.5 Problems ..............................211 7 Finite-Dimensional Vector Spaces 215 7.1 Linear Transformations ......................216 7.2 Inner Product ............................218 7.3 The Determinant ..........................222 7.4 Eigenvectors and Eigenvalues ...................224 7.5 Orthogonal Polynomials ......................227 7.6 Systems of Linear Equations ....................230 7.7 Problems ..............................234 8 Vectors in Relativity 237 8.1 Proper and Coordinate Time ................... 239 8.2 Spacetime Distance ......................... 240 8.3 Lorentz Transformation ...................... 243 8.4 Four-Velocity and Four-Momentum ................ 247 8.4.1 Relativistic Collisions ................... 250 8.4.2 Second Law of Motion ................... 253 8.5 Problems .............................. 254 III Infinite Series 257 9 Infinite Series 259 9.1 Infinite Sequences ..........................259 9.2 Summations .............................262 9.2.1 Mathematical Induction ..................265 9.3 Infinite Series ............................266 9.3.1 Tests for Convergence ...................267 9.3.2 Operations on Series ....................273 9.4 Sequences and Series of Functions ................274 9.4.1 Properties of Uniformly Convergent Series ........277 9.5 Problems ..............................279 10 Application of Common Series 283 10.1 Power Series .............................283 10.1.1 Taylor Series ........................286 10.2 Series for Some Familiar Functions ................287 10.3 Helmholtz Coil ...........................291 10.4 Indeterminate Forms and L Hôpitaľs Rule ............294 xviii________________________________________________________ CONTENTS 10.5 Multipole Expansion........................ 297 10.6 Fourier Series............................ 299 10.7 Multivariable Taylor Series .................... 305 10.8 Application to Differential Equations ............... 307 10.9 Problems .............................. 311 11 Integrals and Series as Functions 317 11.1 Integrals as Functions .......................317 11.1.1 Gamma Function ......................318 11.1.2 The Beta Function .....................320 11.1.3 The Error Function ....................322 11.1.4 Elliptic Functions ......................322 11.2 Power Series as Functions .....................327 11.2.1 Hypergeometric Functions .................328 11.2.2 Confluent Hypergeometric Functions ...........332 11.2.3 Bessel Functions ......................333 11.3 Problems ..............................336 IV Analysis of Vectors 341 12 Vectors and Derivatives 343 12.1 Solid Angle .............................344 12.1.1 Ordinary Angle Revisited .................344 12.1.2 Solid Angle .........................347 12.2 Time Derivative of Vectors ....................350 12.2.1 Equations of Motion in a Central Force Field ......352 12.3 The Gradient ............................355 12.3.1 Gradient and Extrémům Problems ............359 12.4 Problems ..............................362 13 Flux and Divergence 365 13.1 Flux of a Vector Field .......................365 13.1.1 Flux Through an Arbitrary Surface ...........370 13.2 Flux Density = Divergence ....................371 13.2.1 Flux Density ........................371 13.2.2 Divergence Theorem ....................374 13.2.3 Continuity Equation ....................378 13.3 Problems ..............................383 14 Line Integral and Curl 387 14.1 The Line Integral ..........................387 14.2 Curl of a Vector Field and Stokes Theorem ...........391 14.3 Conservative Vector Fields .....................398 14.4 Problems ..............................404 CONTENTS_____________________________________________________________xix 15 Applied Vector Analysis 407 15.1 Double Del Operations ....................... 407 15.2 Magnetic Multipoles ........................ 409 15.3 Laplacian .............................. 411 15.3.1 A Primer of Fluid Dynamics ............... 413 15.4 Maxwell s Equations ........................ 415 15.4.1 Maxwell s Contribution .................. 416 15.4.2 Electromagnetic Waves in Empty Space ......... 417 15.5 Problems .............................. 420 16 Curvilinear Vector Analysis 423 16.1 Elements of Length .........................423 16.2 The Gradient ............................425 16.3 The Divergence ...........................427 16.4 The Curl ..............................431 16.4.1 The Laplacian .......................435 16.5 Problems ..............................436 17 Tensor Analysis 439 17.1 Vectors and Indices .........................439 17.1.1 Transformation Properties of Vectors ...........441 17.1.2 Covariant and Contravariant Vectors ...........445 17.2 From Vectors to Tensors ......................447 17.2.1 Algebraic Properties of Tensors ..............450 17.2.2 Numerical Tensors .....................452 17.3 Metric Tensor ............................454 17.3.1 Index Raising and Lowering ................457 17.3.2 Tensors and Electrodynamics ...............459 17.4 Differentiation of Tensors .....................462 17.4.1 Covariant Differential and Affine Connection ......462 17.4.2 Covariant Derivative ....................464 17.4.3 Metric Connection .....................465 17.5 Riemann Curvature Tensor ....................468 17.6 Problems ..............................471 V Complex Analysis 475 18 Complex Arithmetic 477 18.1 Cartesian Form of Complex Numbers ...............477 18.2 Polar Form of Complex Numbers .................482 18.3 Fourier Series Revisited ......................488 18.4 A Representation of Delta Function ...............491 18.5 Problems ..............................493 xx______________________________________________________________ CONTENTS 19 Complex Derivative and Integral 497 19.1 Complex Functions ......................... 497 19.1.1 Derivatives of Complex Functions ............. 499 19.1.2 Integration of Complex Functions ............. 503 19.1.3 Cauchy Integral Formula ................. 508 19.1.4 Derivatives as Integrals .................. 509 19.2 Problems .............................. 511 20 Complex Series 515 20.1 Power Series .............................516 20.2 Taylor and Laurent Series .....................518 20.3 Problems ..............................522 21 Calculus of Residues 525 21.1 The Residue .............................525 21.2 Integrals of Rational Functions ..................529 21.3 Products of Rational and Trigonometric Functions .......532 21.4 Functions of Trigonometric Functions ..............534 21.5 Problems ..............................536 VI Differential Equations 539 22 From PDEs to ODEs 541 22.1 Separation of Variables .......................542 22.2 Separation in Cartesian Coordinates ...............544 22.3 Separation in Cylindrical Coordinates ..............547 22.4 Separation in Spherical Coordinates ...............548 22.5 Problems ..............................550 23 First-Order Differential Equations 551 23.1 Normal Form of a FODE .....................551 23.2 Integrating Factors .........................553 23.3 First-Order Linear Differential Equations ............556 23.4 Problems ..............................561 24 Second-Order Linear Differential Equations 563 24.1 Linearity, Superposition, and Uniqueness .............564 24.2 TheWronskian ...........................566 24.3 A Second Solution to the HSOLDE ................567 24.4 The General Solution to an ISOLDE ...............569 24.5 Sturm-Liouville Theory ......................570 24.5.1 Adjoint Differential Operators ..............571 24.5.2 Sturm-Liouville System ..................574 24.6 SOLDEs with Constant Coefficients ...............575 24.6.1 The Homogeneous Case ..................576 24.6.2 Central Force Problem ...................579 CONTENTS 24.6.3 The Inhomogeneous Case .................583 24.7 Problems..............................587 25 Laplace s Equation: Cartesian Coordinates 591 25.1 Uniqueness of Solutions ......................592 25.2 Cartesian Coordinates .......................594 25.3 Problems ..............................603 26 Laplace s Equation: Spherical Coordinates 607 26.1 Erobenius Method .........................608 26.2 Legendre Polynomials .......................610 26.3 Second Solution of the Legendre DE...............617 26.4 Complete Solution .........................619 26.5 Properties of Legendre Polynomials ................622 26.5.1 Parity ............................622 26.5.2 Recurrence Relation ....................622 26.5.3 Orthogonality ........................624 26.5.4 Rodrigues Formula .....................626 26.6 Expansions in Legendre Polynomials ...............628 26.7 Physical Examples .........................631 26.8 Problems ..............................635 27 Laplace s Equation: Cylindrical Coordinates 639 27.1 The ODEs .............................. 639 27.2 Solutions of the Bessel DE..................... 642 27.3 Second Solution of the Bessel DE................. 645 27.4 Properties of the Bessel Functions ................ 646 27.4.1 Negative Integer Order ................... 646 27.4.2 Recurrence Relations .................... 646 27.4.3 Orthogonality ........................ 647 27.4.4 Generating Function .................... 649 27.5 Expansions in Bessel Functions .................. 653 27.6 Physical Examples ......................... 654 27.7 Problems .............................. 657 28 Other PDEs of Mathematical Physics 661 28.1 The Heat Equation .........................661 28.1.1 Heat-Conducting Rod ...................662 28.1.2 Heat Conduction in a Rectangular Plate .........663 28.1.3 Heat Conduction in a Circular Plate ...........664 28.2 The Schrödinger Equation .....................666 28.2.1 Quantum Harmonic Oscillator ..............667 28.2.2 Quantum Particle in a Box ................675 28.2.3 Hydrogen Atom ......................677 28.3 The Wave Equation ........................680 28.3.1 Guided Waves .......................682 XXII CONTENTS 28.3.2 Vibrating Membrane ....................686 28.4 Problems ..............................687 VII Special Topics 691 29 Integral Transforms 693 29.1 The Fourier Transform .......................693 29.1.1 Properties of Fourier Transform ..............696 29.1.2 Sine and Cosine Transforms ................697 29.1.3 Examples of Fourier Transform ..............698 29.1.4 Application to Differential Equations ...........702 29.2 Fourier Transform and Green s Functions ............705 29.2.1 Green s Function for the Laplacian ............708 29.2.2 Green s Function for the Heat Equation .........709 29.2.3 Green s Function for the Wave Equation .........711 29.3 The Laplace Transform ......................712 29.3.1 Properties of Laplace Transform .............713 29.3.2 Derivative and Integral of the Laplace Transform .... 717 29.3.3 Laplace Transform and Differential Equations ......718 29.3.4 Inverse of Laplace Transform ...............721 29.4 Problems ..............................723 30 Calculus of Variations 727 30.1 Variational Problem ........................728 30.1.1 Euler-Lagrange Equation .................729 30.1.2 Beltrami identity ......................731 30.1.3 Several Dependent Variables ...............734 30.1.4 Several Independent Variables ...............734 30.1.5 Second Variation ......................735 30.1.6 Variational Problems with Constraints ..........738 30.2 Lagrangian Dynamics .......................740 30.2.1 From Newton to Lagrange.................740 30.2.2 Lagrangian Densities ....................744 30.3 Hamiltonian Dynamics .......................747 30.4 Problems ..............................750 31 Nonlinear Dynamics and Chaos 753 31.1 Systems Obeying Iterated Maps ..................754 31.1.1 Stable and Unstable Fixed Points .............755 31.1.2 Bifurcation .........................757 31.1.3 Onset of Chaos .......................761 31.2 Systems Obeying DEs .......................763 31.2.1 The Phase Space ......................764 31.2.2 Autonomous Systems ...................766 31.2.3 Onset of Chaos .......................770 CONTENTS____________________________________________________________xxiii 31.3 Universality of Chaos........................773 31.3.1 Feigenbaum Numbers ...................773 31.3.2 Fractal Dimension.....................775 31.4 Problems ..............................778 32 Probability Theory 781 32.1 Basic Concepts ...........................781 32.1.1 A Set Theory Primer ....................782 32.1.2 Sample Space and Probability ...............784 32.1.3 Conditional and Marginal Probabilities .........786 32.1.4 Average and Standard Deviation .............789 32.1.5 Counting: Permutations and Combinations .......791 32.2 Binomial Probability Distribution .................792 32.3 Poisson Distribution ........................797 32.4 Continuous Random Variable ...................801 32.4.1 Transformation of Variables ................804 32.4.2 Normal Distribution ....................806 32.5 Problems ..............................809 Bibliography 815 Index 817
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spelling	Hassani, Sadri Verfasser (DE-588)114976838X aut Mathematical methods for students of physics and related fields Sadri Hassani 2. ed. New York [u.a.] Springer 2009 XXIII, 829 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematische Physik Mathematical physics Problems, exercises, etc Mathematical physics Study and teaching Topologie (DE-588)4060425-1 gnd rswk-swf Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Kategorie Mathematik (DE-588)4129930-9 gnd rswk-swf (DE-588)1071861417 Konferenzschrift 1978 Berlin West gnd-content Mathematische Physik (DE-588)4037952-8 s DE-604 Kategorie Mathematik (DE-588)4129930-9 s Topologie (DE-588)4060425-1 s Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018623775&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Hassani, Sadri Mathematical methods for students of physics and related fields Mathematische Physik Mathematical physics Problems, exercises, etc Mathematical physics Study and teaching Topologie (DE-588)4060425-1 gnd Mathematische Physik (DE-588)4037952-8 gnd Kategorie Mathematik (DE-588)4129930-9 gnd
subject_GND	(DE-588)4060425-1 (DE-588)4037952-8 (DE-588)4129930-9 (DE-588)1071861417
title	Mathematical methods for students of physics and related fields
title_auth	Mathematical methods for students of physics and related fields
title_exact_search	Mathematical methods for students of physics and related fields
title_full	Mathematical methods for students of physics and related fields Sadri Hassani
title_fullStr	Mathematical methods for students of physics and related fields Sadri Hassani
title_full_unstemmed	Mathematical methods for students of physics and related fields Sadri Hassani
title_short	Mathematical methods
title_sort	mathematical methods for students of physics and related fields
title_sub	for students of physics and related fields
topic	Mathematische Physik Mathematical physics Problems, exercises, etc Mathematical physics Study and teaching Topologie (DE-588)4060425-1 gnd Mathematische Physik (DE-588)4037952-8 gnd Kategorie Mathematik (DE-588)4129930-9 gnd
topic_facet	Mathematische Physik Mathematical physics Problems, exercises, etc Mathematical physics Study and teaching Topologie Kategorie Mathematik Konferenzschrift 1978 Berlin West
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018623775&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT hassanisadri mathematicalmethodsforstudentsofphysicsandrelatedfields

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