Mathematical methods in the physical sciences
Boas, Mary L.
Mathematical methods in the physical sciences - Ed. 3 - New Delhi John Wiley Sons, Inc. 2009 - xix,839p.
CONTENTS 1INFINITE SERIES, POWER SERIES 1 1.The Geometric Series1 2.Definitions and Notation4 3.Applications of Series6 4.Convergent and Divergent Series 6 5.Testing Series for Convergence; the Preliminary Test9 6.Convergence Tests for Series of Positive Terms: Absolute Convergence 10 A. The Corrparison Test10 B. The Integral Test11 C. The Ratio Test13 D. A Special Comparison Test15 7.Alternating Series17 8.Conditionally Convergent Series18 9.Useful Facts About Series19 10.Power Series; Interval of Convergence20 11.Theorems About Power Series 23 12.Expanding Functions in Power Series 23 13.Techniques for Obtaining Power Series Expansions25 A. Multiplying a Series by a Polynomial or by Another Series26 B. Division of Two Series or of a Series by a Polynomial27 C. Binomial Series28 D. Substitution of a Polynomial or a Series for the Variable in Another Series 29 E. Combination of Methods30 F. Taylor Series Using the Basic Maclaurin Series 30 G. Using a Computer 31 14.Accuracy of Series Approximations 33 15.Some Uses of Series36 16.Miscellaneous Problems 44 2COMPLEX NUMBERS 46 1.Introduction46 2.Real and Imaginary Parts of a Complex Number 47 3.The Complex Plane 47 4.Terminology and Notation49 5.Complex Algebra51 A. Simplifying to x+iy form 51 B. Complex Conjugate of a Complex Expression52 C. Finding the Absolute Value of z 53 D. Complex Equations54 E. Graphs 54 F. Physical Applications 55 6.Complex Infinite Series 56 7.Complex Power Series; Disk of Convergence 58 8.Elementary Functions of Complex Numbers 60 9.Euler's Formula 61 10.Powers and Roots of Complex Numbers64 11.The Exponential and Trigonometric Functions67 12.Hyperbolic Functions 70 13.Logarithms72 14.Complex Roots and Powers 73 15.Inverse Trigonometric and Hyperbolic Functions74 16.Some Applications76 17.Miscellaneous Problems 80 3LINEAR ALGEBRA 82 1. Introduction 82 2. Matrices; Row Reduction83 3. Determinants; Cramer's Rule89 4. Vectors96 5. Lines and Planes106 6. Matrix Operations114 7. Linear Combinations, Linear Functions, Linear Operators124 8. Linear Dependence and Independence132 9. Special Matrices and Formulas137 10. Linear Vector Spaces142 11. Eigenvalues and Eigenvectors; Diagonalizing Matrices148 12. Applications of Diagonalization162 13. A Brief Introduction to Groups172 14. General Vector Spaces 179 15. Miscellaneous Problems184 4 PARTIAL DIFFERENTIATION 188 1. Introduction and Notation188 2. Power Series in Two Variables191 3. Total Differentials193 4. Approximations using Differentials196 5. Chain Rule or Differentiating a Function of a. Function199 6. Implicit Differentiation202 7. More Chain Rule 203 8. Application of Partial Differentiation to Maximum and Minimum Problems 211 9. Maximum and Minimum Problems with Constraints; Lagrange Multipliers 214 10. Endpoint or Boundary Point Problems 223 11. Change of Variables228 12. Differentiation of Integrals; Leibniz' Rule 233 13. Miscellaneous problems238 5 MULTIPLE INTEGRALS 241 1. Introduction 241 2. Double and Triple Integrals242 3. Applications of Integration; Single and Multiple Integrals249 4. Change of Variables in Integrals; Jacobians 258 5. Surface Integrals 270 6. Miscellaneous Problems273 6 VECTOR ANALYSIS 276 1. Introduction276 2. Applications of Vector Multiplication276 3. Triple Products 278 4. Differentiation of Vectors 285 5. Fields 289 6. Directional Derivative; Gradient290 7. Some Other Expressions Involving V296 8. Line Integrals299 9. Green's Theorem in the Plane309 10. The Divergence and the Divergence Theorem314 11. The Curl and Stokes' Theorem324 12. Miscellaneous Problems 336 7 FOURIER SERIES AND TRANSFORMS 340 1. Introduction340 2. Simple Harmonic Motion and Wave Motion; Periodic Functions340 3. Applications of Fourier Series345 4. Average Value of a Function 347 5. Fourier Coefficients 350 6. Dirichlet Conditions355 7. Complex Form of Fourier Series 358 8. Other Intervals360 9. Even and Odd Functions364 10. An Application to Sound372 11. Parseval's Theorem 375 12. Fourier Transforms378 13. Miscellaneous Problems 386 8ORDINARY DIFFERENTIAL EQUATIONS 39O 1. Introduction390 2. Separable Equations395 3. Linear First-Order Equations401 4. Other Methods for First-Order Equations 404 5. Second-Order Linear Equations with Constant Coefficients and Zero Right-Hand Side 408 6. Second-Order Linear Equations with Constant Coefficients and Right-Hand Side Not Zero417 7. Other Second-Order Equations430 8. The Laplace Transform437 9. Solution of Differential Equations by Laplace Transforms440 10. Convolution444 11. The Dirac Delta Function449 12. A Brief Introduction to Green Functions461 13. Miscellaneous Problems466 9CALCULUS OF VARIATIONS 472 1. Introduction472 2. The Euler Equation474 3. Using the Euler Equation473 4. The Brachistochrone Problem; Cycloids482 5. Several Dependent Variables; Lagrange's Equations485 6. Isoperimetric Problems491 7. Variational Notation493 8. Miscellaneous Problems494 10 TENSOR ANALYSIS 496 1. Introduction496 2. Cartesian Tensors498 3.. Tensor Notation and Operations502 4. Inertia Tensor505 5. Kronecker Delta and Levi-Civita Symbol508 6. Pseudovectors and Pseudotensors514 7. More About Applications518 8. Curvilinear Coordinates 521 9. Vector Operators in Orthogonal Curvilinear Coordinates525 10. Non-Cartesian Tensors529 11. Miscellaneous Problems535 11 SPECIAL FUNCTIONS 537 1. Introduction537 2. The Factorial Function 538 3. Definition of the Gamma Function; Recursion Relation538 4. The Gamma Function of Negative Numbers540 5. Some Important Formulas Involving Gamma Functions 541 6. Beta Functions 542 7. Beta Functions in Terms of Gamma Functions 543 8. The Simple Pendulum545 9. The Error Function 547 10. Asymptotic Series549 11. Stirling's Formula552 12. Elliptic Integrals and Functions554 13. Miscellaneous Problems 560 12 SERIES SOLUTIONS OF DIFFERENTIAL EQUATIONS; LEGENDRE, BESSEL, HERMITE, AND LAGUERRE FUNCTIONS 562 1. Introduction562 2. Legendre's Equation564 3. Leibniz' Rule for Differentiating Products567 4. Rodrigues' Formula568 5. Generating Function for Legendre Polynomials 569 6. Complete Sets of Orthogonal Functions575 7. Orthogonality of the Legendre Polynomials577 8. Normalization of the Legendre Polynomials578 9. Legendre Series580 10. The Associated Legendre Functions583 11. Generalized Power Series or the Method of Frobenius 585 12. Bessel's Equation587 13. The Second Solution of Bessel's Equation 590 14. Graphs and Zeros of Bessel Functions591 15. Recursion Relations592 16. Differential Equations with Bessel Function Solutions593 17. Other Kinds of Bessel Functions 595 18. The Lengthening Pendulum598 19. Orthogonality of Bessel Functions601 20. Approximate Formulas for Bessel Functions604 21. Series Solutions; Fuchs's Theorem 605 22. Hermite Functions; Laguerre Functions; Ladder Operators607 23. Miscellaneous Problems 615 13 PARTIAL DIFFERENTIAL EQUATIONS 619 1. Introduction619 2. Laplace's Equation; Steady-State Temperature in a Rectangular Plate 621 3. The Diffusion or Heat Flow Equation; the Schrodinger Equation628 4. The Wave Equation; the Vibrating String633 5. Steady-state Temperature in a Cylinder638 6. Vibration of a Circular Membrane644 7. Steady-state Temperature in a Sphere647 8. Poisson's Equation652 9. Integral Transform Solutions of Partial Differential Equations659 10. Miscellaneous Problems663 14 FUNCTIONS OF A COMPLEX VARIABLE666 1. Introduction 666 2. Analytic Functions667 3. Contour Integrals674 4. Laurent Series678 5. The Residue Theorem682 6. Methods of Finding Residues683 7. Evaluation of Definite Integrals by Use of the Residue Theorem 687 8. The Point at Infinity; Residues at Infinity702 9. Mapping705 10. Some Applications of Conformal Mapping 710 11. Miscellaneous Problems 718 15 PROBABILITY AND STATISTICS 722 1. Introduction722 2. Sample Space724 3. Probability Theorems729 4. Methods of Counting736 5. Random Variables 744 6. Continuous Distributions750 7. Binomial Distribution 756 8. The Normal or Gaussian Distribution761 9. The Poisson Distribution 767 10. Statistics and Experimental Measurements 770 11. Miscellaneous Problems776 REFERENCES 779 ANSWERS TO SELECTED PROBLEMS 781 INDEX 811
8126508108
510 / BOA
Mathematical methods in the physical sciences - Ed. 3 - New Delhi John Wiley Sons, Inc. 2009 - xix,839p.
CONTENTS 1INFINITE SERIES, POWER SERIES 1 1.The Geometric Series1 2.Definitions and Notation4 3.Applications of Series6 4.Convergent and Divergent Series 6 5.Testing Series for Convergence; the Preliminary Test9 6.Convergence Tests for Series of Positive Terms: Absolute Convergence 10 A. The Corrparison Test10 B. The Integral Test11 C. The Ratio Test13 D. A Special Comparison Test15 7.Alternating Series17 8.Conditionally Convergent Series18 9.Useful Facts About Series19 10.Power Series; Interval of Convergence20 11.Theorems About Power Series 23 12.Expanding Functions in Power Series 23 13.Techniques for Obtaining Power Series Expansions25 A. Multiplying a Series by a Polynomial or by Another Series26 B. Division of Two Series or of a Series by a Polynomial27 C. Binomial Series28 D. Substitution of a Polynomial or a Series for the Variable in Another Series 29 E. Combination of Methods30 F. Taylor Series Using the Basic Maclaurin Series 30 G. Using a Computer 31 14.Accuracy of Series Approximations 33 15.Some Uses of Series36 16.Miscellaneous Problems 44 2COMPLEX NUMBERS 46 1.Introduction46 2.Real and Imaginary Parts of a Complex Number 47 3.The Complex Plane 47 4.Terminology and Notation49 5.Complex Algebra51 A. Simplifying to x+iy form 51 B. Complex Conjugate of a Complex Expression52 C. Finding the Absolute Value of z 53 D. Complex Equations54 E. Graphs 54 F. Physical Applications 55 6.Complex Infinite Series 56 7.Complex Power Series; Disk of Convergence 58 8.Elementary Functions of Complex Numbers 60 9.Euler's Formula 61 10.Powers and Roots of Complex Numbers64 11.The Exponential and Trigonometric Functions67 12.Hyperbolic Functions 70 13.Logarithms72 14.Complex Roots and Powers 73 15.Inverse Trigonometric and Hyperbolic Functions74 16.Some Applications76 17.Miscellaneous Problems 80 3LINEAR ALGEBRA 82 1. Introduction 82 2. Matrices; Row Reduction83 3. Determinants; Cramer's Rule89 4. Vectors96 5. Lines and Planes106 6. Matrix Operations114 7. Linear Combinations, Linear Functions, Linear Operators124 8. Linear Dependence and Independence132 9. Special Matrices and Formulas137 10. Linear Vector Spaces142 11. Eigenvalues and Eigenvectors; Diagonalizing Matrices148 12. Applications of Diagonalization162 13. A Brief Introduction to Groups172 14. General Vector Spaces 179 15. Miscellaneous Problems184 4 PARTIAL DIFFERENTIATION 188 1. Introduction and Notation188 2. Power Series in Two Variables191 3. Total Differentials193 4. Approximations using Differentials196 5. Chain Rule or Differentiating a Function of a. Function199 6. Implicit Differentiation202 7. More Chain Rule 203 8. Application of Partial Differentiation to Maximum and Minimum Problems 211 9. Maximum and Minimum Problems with Constraints; Lagrange Multipliers 214 10. Endpoint or Boundary Point Problems 223 11. Change of Variables228 12. Differentiation of Integrals; Leibniz' Rule 233 13. Miscellaneous problems238 5 MULTIPLE INTEGRALS 241 1. Introduction 241 2. Double and Triple Integrals242 3. Applications of Integration; Single and Multiple Integrals249 4. Change of Variables in Integrals; Jacobians 258 5. Surface Integrals 270 6. Miscellaneous Problems273 6 VECTOR ANALYSIS 276 1. Introduction276 2. Applications of Vector Multiplication276 3. Triple Products 278 4. Differentiation of Vectors 285 5. Fields 289 6. Directional Derivative; Gradient290 7. Some Other Expressions Involving V296 8. Line Integrals299 9. Green's Theorem in the Plane309 10. The Divergence and the Divergence Theorem314 11. The Curl and Stokes' Theorem324 12. Miscellaneous Problems 336 7 FOURIER SERIES AND TRANSFORMS 340 1. Introduction340 2. Simple Harmonic Motion and Wave Motion; Periodic Functions340 3. Applications of Fourier Series345 4. Average Value of a Function 347 5. Fourier Coefficients 350 6. Dirichlet Conditions355 7. Complex Form of Fourier Series 358 8. Other Intervals360 9. Even and Odd Functions364 10. An Application to Sound372 11. Parseval's Theorem 375 12. Fourier Transforms378 13. Miscellaneous Problems 386 8ORDINARY DIFFERENTIAL EQUATIONS 39O 1. Introduction390 2. Separable Equations395 3. Linear First-Order Equations401 4. Other Methods for First-Order Equations 404 5. Second-Order Linear Equations with Constant Coefficients and Zero Right-Hand Side 408 6. Second-Order Linear Equations with Constant Coefficients and Right-Hand Side Not Zero417 7. Other Second-Order Equations430 8. The Laplace Transform437 9. Solution of Differential Equations by Laplace Transforms440 10. Convolution444 11. The Dirac Delta Function449 12. A Brief Introduction to Green Functions461 13. Miscellaneous Problems466 9CALCULUS OF VARIATIONS 472 1. Introduction472 2. The Euler Equation474 3. Using the Euler Equation473 4. The Brachistochrone Problem; Cycloids482 5. Several Dependent Variables; Lagrange's Equations485 6. Isoperimetric Problems491 7. Variational Notation493 8. Miscellaneous Problems494 10 TENSOR ANALYSIS 496 1. Introduction496 2. Cartesian Tensors498 3.. Tensor Notation and Operations502 4. Inertia Tensor505 5. Kronecker Delta and Levi-Civita Symbol508 6. Pseudovectors and Pseudotensors514 7. More About Applications518 8. Curvilinear Coordinates 521 9. Vector Operators in Orthogonal Curvilinear Coordinates525 10. Non-Cartesian Tensors529 11. Miscellaneous Problems535 11 SPECIAL FUNCTIONS 537 1. Introduction537 2. The Factorial Function 538 3. Definition of the Gamma Function; Recursion Relation538 4. The Gamma Function of Negative Numbers540 5. Some Important Formulas Involving Gamma Functions 541 6. Beta Functions 542 7. Beta Functions in Terms of Gamma Functions 543 8. The Simple Pendulum545 9. The Error Function 547 10. Asymptotic Series549 11. Stirling's Formula552 12. Elliptic Integrals and Functions554 13. Miscellaneous Problems 560 12 SERIES SOLUTIONS OF DIFFERENTIAL EQUATIONS; LEGENDRE, BESSEL, HERMITE, AND LAGUERRE FUNCTIONS 562 1. Introduction562 2. Legendre's Equation564 3. Leibniz' Rule for Differentiating Products567 4. Rodrigues' Formula568 5. Generating Function for Legendre Polynomials 569 6. Complete Sets of Orthogonal Functions575 7. Orthogonality of the Legendre Polynomials577 8. Normalization of the Legendre Polynomials578 9. Legendre Series580 10. The Associated Legendre Functions583 11. Generalized Power Series or the Method of Frobenius 585 12. Bessel's Equation587 13. The Second Solution of Bessel's Equation 590 14. Graphs and Zeros of Bessel Functions591 15. Recursion Relations592 16. Differential Equations with Bessel Function Solutions593 17. Other Kinds of Bessel Functions 595 18. The Lengthening Pendulum598 19. Orthogonality of Bessel Functions601 20. Approximate Formulas for Bessel Functions604 21. Series Solutions; Fuchs's Theorem 605 22. Hermite Functions; Laguerre Functions; Ladder Operators607 23. Miscellaneous Problems 615 13 PARTIAL DIFFERENTIAL EQUATIONS 619 1. Introduction619 2. Laplace's Equation; Steady-State Temperature in a Rectangular Plate 621 3. The Diffusion or Heat Flow Equation; the Schrodinger Equation628 4. The Wave Equation; the Vibrating String633 5. Steady-state Temperature in a Cylinder638 6. Vibration of a Circular Membrane644 7. Steady-state Temperature in a Sphere647 8. Poisson's Equation652 9. Integral Transform Solutions of Partial Differential Equations659 10. Miscellaneous Problems663 14 FUNCTIONS OF A COMPLEX VARIABLE666 1. Introduction 666 2. Analytic Functions667 3. Contour Integrals674 4. Laurent Series678 5. The Residue Theorem682 6. Methods of Finding Residues683 7. Evaluation of Definite Integrals by Use of the Residue Theorem 687 8. The Point at Infinity; Residues at Infinity702 9. Mapping705 10. Some Applications of Conformal Mapping 710 11. Miscellaneous Problems 718 15 PROBABILITY AND STATISTICS 722 1. Introduction722 2. Sample Space724 3. Probability Theorems729 4. Methods of Counting736 5. Random Variables 744 6. Continuous Distributions750 7. Binomial Distribution 756 8. The Normal or Gaussian Distribution761 9. The Poisson Distribution 767 10. Statistics and Experimental Measurements 770 11. Miscellaneous Problems776 REFERENCES 779 ANSWERS TO SELECTED PROBLEMS 781 INDEX 811
8126508108
510 / BOA