Advanced engineering mathematics / Erwin Kreyszig.
Editor: New York : Wiley, c1967Edición: 2nd edDescripción: xvii, 898 p. : il. ; 25 cmOtra clasificación: 00A06CONTENTS Introduction Review of Some Topics From Algebra and Calculus [1] 0.1 Elementary Functions, [1] 0.2 Partial Derivatives, [8] 0.3 Second- and Third-Order Determinants, [10] 0.4 Complex Numbers, [18] 0.5 Polar Form of Complex Numbers, [22] 0.6 Some General Remarks About Numerical Computations, [24] 0.7 Solution of Equations, [26] 0.8 Approximate Integration, [31] Chapter 1 Ordinary Differential Equations of the First Order [39] 1.1 Basic Concepts and Ideas, [39] 1.2 Geometrical Considerations. Isoclines, [46] 1.3 Separable Equations, [49] 1.4 Equations Reducible to Separable Form, [57] 1.5 Exact Differential Equations, [59] 1.6 Integrating Factors, [61] 1.7 Linear First-Order Differential Equations, [63] 1.8 Variation of Parameters, [67] 1.9 Electric Circuits, [69] 1.10 Families of Curves. Orthogonal Trajectories, [74] 1.11 Picard’s Iteration Method, [79] 1.12 Existence and Uniqueness of Solutions, [82] 1.13 Numerical Methods for Differential Equations of the First Order, [86] Chapter 2 Ordinary Linear Differential Equations [93] 2.1 Homogeneous Linear Equations of the Second Order, [94] 2.2 Homogeneous Second-Order Equations with Constant Coefficients, [97] 2.3 General Solution. Fundamental System, [99] 2.4 Complex Roots of the Characteristic Equation. Initial Value Problem, [102] 2.5 Double Root of the Characteristic Equation, [106] 2.6 Free Oscillations, [108] 2.7 Cauchy Equation, [116] 2.8 Existence and Uniqueness of Solutions, [117] 2.9 Homogeneous Linear Equations of Arbitrary Order, [123] 2.10 Homogeneous Linear Equations of Arbitrary Order with Constant Coefficients, [126] 2.11 Nonhomogeneous Linear Equations, [128] 2.12 A Method for Solving Nonhomogeneous Linear Equations, [130] 2.13 Forced Oscillations. Resonance, [134] 2.14 Electric Circuits, [140] 2.15 Complex Method for Obtaining Particular Solutions, [144] 2.16 General Method for Solving Nonhomogeneous Equations, [147] 2.17 Numerical Methods for Second-Order Differential Equations, [149] Chapter 3 Power Series Solutions of Differential Equations [155] 3.1 The Power Series Method, [155] 3.‘2 Theoretical Basis of the Power Series Method, [159] 3.3 Legendre’s Equation. Legendre Polynomials, [164] 3.4 Extended Power Series Method. Indicial Equation, [168] 3.5 Bessel’s Equation. Bessel Functions of the First Kind, [179] 3.6 Further Properties of Bessel Functions of the First Kind, [185] 3.7 Bessel Functions of the Second Kind, [187] Chapter 4 Laplace Transformation [192] 4.1 Laplace Transform. Inverse Transform. Linearity, [192] 4.2 Laplace Transforms of Derivatives and Integrals, [198] 4.3 Transformation of Ordinary Differential Equations, [201] 4.4 Partial Fractions, [204] 4.5 Examples and Applications, [209] 4.6 Differentiation and Integration of Transforms, [213] 4.7 Unit Step Function, [216] 4.8 Shifting on the t-axis, [219] 4.9 Periodic Functions, [223] 4.10 Table 17. Some Laplace Transforms, [231] Chapter 5 Vector Analysis [235] 5.1 Scalars and Vectors, [235] 5.2 Components of a Vector, [237] 5.3 Vector Addition. Multiplication by Scalars, [239] 5.4 Scalar Product, [242] 5.5 Vector Product, [247] 5.6 Vector Products in Terms of Components, [248] 5.7 Scalar Triple Product. Linear Dependence of Vectors, [254] 5.8 Other Repeated Products, [258] 5.9 Scalar Fields and Vector Fields, [260] 5.10 Vector Calculus, [263] 5.11 Curves, [265] 5.12 Arc Length, [268] 5.13 Tangent. Curvature and Torsion, [270] 5.14 Velocity and Acceleration, [274] 5.15 Chain Rule and Mean Value Theorem for Functions of Several Variables, [278] 5.16 Directional Derivative. Gradient of a Scalar Field, [282] 5.17 Transformation of Coordinate Systems and Vector Components, [288] 5.18 Divergence of a Vector Field, [292] 5.19 Curl of a Vector Field, [296] Chapter 6 Line and Surface Integrals. Integral Theorems [299] 6.1 Line Integral, [299] 6.2 Evaluation of Line Integrals, [302] 6.3 Double Integrals, [306] 6.4 Transformation of Double Integrals into Line Integrals, [313] 6.5 Surfaces, [318] 6.6 Tangent Plane. First Fundamental Form, Area, [321] 6.7 Surface Integrals, [327] 6.8 Triple Integrals. Divergence Theorem of Gauss, [332] 6.9 Consequences and Applications of the Divergence Theorem, [336] 6.10 Stokes’s Theorem, [342] 6.11 Consequences and Applications of Stokes’s Theorem, [346] 6.12 Line Integrals Independent of Path, [347] Chapter 7 Matrices and Determinants. Systems of Linear Equations [356] 7.1 Basic Concepts. Addition of Matrices, [356] 7.2 Matrix Multiplication, [361] 7.3 Determinants, [368] 7.4 Submatrices. Rank, [380] 7.5 Systems of n Linear Equations in n Unknowns. Cramer’s Rule, [382] 7.6 Arbitrary Homogeneous Systems of Linear Equations, [386] 7.7 Arbitrary Nonhomogeneous Systems of Linear Equations, [392] 7.8 Further Properties of Systems of Linear Equations, [395] 7.9 Gauss’s Elimination Method, [398] 7.10 The Inverse of a Matrix, [401] 7.11 Eigenvalues. Eigenvectors, [406] 7.12 Bilinear, Quadratic, Hermitian, and Skew-Hermitian Forms, [413] 7.13 Eigenvalues of Hermitian, Skew-Hermitian, and Unitary Matrices, [417] 7.14 Bounds for Eigenvalues, [422] 7.15 Determination of Eigenvalues by Iteration, [426] Chapter 8 Fourier Series and Integrals [431] 8.1 Periodic Functions. Trigonometric Series, [431] 8.2 Fourier Series. Euler’s Formulas, [434] 8.3 Even and Odd Functions, [440] 8.4 Functions Having Arbitrary Period, [444] 8.5 Half-Range Expansions, [447] 8.6 Determination of Fourier Coefficients Without Integration, [451] 8.7 Forced Oscillations, [455] 8.8 Numerical Methods for Determining Fourier Coefficients. Square Error, [458] 8.9 Instrumental Methods for Determining Fourier Coefficients, [464] 8.10 The Fourier Integral, [465] 8.11 Orthogonal Functions, [473] 8.12 Sturm-Liouville Problem, [476] 8.13 Orthogonality of Bessel Functions, [483] Chapter 9 Partial Differential Equations [486] 9.1 Basic Concepts, [486] 9.2 Vibrating String. One-Dimensional Wave Equation, [488] 9.3 Separation of Variables (Product Method), [490] 9.4 D’Alembert’s Solution of the Wave Equation, [499] 9.5 One-Dimensional Heat Flow, [501] 9.6 Heat Flow in an Infinite Bar, [506] 9.7 Vibrating Membrane. Two-Dimensional Wave Equation, [510] 9.8 Rectangular Membrane, [512] 9.9 Laplacian in Polar Coordinates, [519] 9.10 Circular Membrane. Bessel’s Equation, [521] 9.11 Laplace’s Equation. Potential, [526] 9.12 Laplace’s Equation in Spherical Coordinates. Legendre’s Equation, [529] Chapter 10 Complex Analytic Functions [534] 10.1 Complex Numbers. Triangle Inequality, [535] 10.2 Limit. Derivative. Analytic Function, [539] 10.3 Cauchy*Riemann Equations. Laplace’s Equation, [543] 10.4 Rational Functions. Root, [547] 10.5 Exponential Function, [550] 10.6 Trigonometric and Hyperbolic Functions, [553] 10.7 Logarithm. General Power, [556] Chapter 11 Conformal Mapping [560] 11.1 Mapping, [560] 11.2 Conformal Mapping, [564] 11.3 Linear Transformations, [568] 11.4 Special Linear Transformations, [572] 11.5 Mapping by Other Elementary Functions, [577] 11.6 Riemann Surfaces, [584] Chapter 12 Complex Integrals [588] 12.1 Line Integral in the Complex Plane, [588] 12.2 Basic Properties of the Complex Line Integral, [594] 12.3 Cauchy’s Integral Theorem, [595] 12.4 Evaluation of Line Integrals by Indefinite Integration, [602] 12.5 Cauchy’s Integral Formula, [605] 12.6 The Derivatives of an Analytic Function, [607] Chapter 13 Sequences and Series [611] 13.1 Sequences, [611] 13.2 Series, [618] 13.3 Tests for Convergence and Divergence of Series, [623] 13.4 Operations on Series, [629] 13.5 Power Series, [633] 13.6 Functions Represented by Power Series, [640] Chapter 14 Taylor and Laurent Series [646] 14.1 Taylor Series, [646] 14.2 Taylor Series of Elementary Functions, [650] 14.3 Practical Methods for Obtaining Power Series, [652] 14.4 Uniform Convergence, [655] 14.5 Laurent Series, [663] 14.6 Behavior of Functions at Infinity, [668] Chapter 15 Integration by the Method of Residues [671] 15.1 Zeros and Singularities, [671] 15.2 Residues, [675] 15.3 The Residue Theorem, [678] 15.4 Evaluation of Real Integrals, [680] Chapter 16 Complex Analytic Functions and Potential Theory [689] 16.1 Electrostatic Fields, [689] 16.2 Two-Dimensional Fluid How, [693] 16.3 Special Complex Potentials, [697] 16.4 General Properties of Harmonic Functions, [702] 16.5 Poisson’s Integral Formula, [705] Chapter 17 Special Functions. Asymptotic Expansions [710] VIA Gamma and Beta Functions, [710] 17.2 Error Function. Fresnel Integrals. Sine and Cosine Integrals, [716] 17.3 Asymptotic Expansions, [720] 17.4 Further Properties of Asymptotic Expansions, [725] Chapter 18 Probability and Statistics [732] 18.1 Nature and Purpose of Mathematical Statistics, [732] 18.2 Tabular and Graphical Representation of Samples, [734] 18.3 Sample Mean and Sample Variance, [740] 18.4 Random Experiments, Outcomes, Events, [744] 18.5 Probability, [748] 18.6 Permutations and Combinations, [752] 18.7 Random Variables. Discrete and Continuous Distributions, [756] 18.8 Mean and Variance of a Distribution, [762] 18.9 Binomial, Poisson, and Hypergeometric Distributions, [766] 18.10 Normal Distribution, [770] 18.11 Distributions of Several Random Variables, [777] 18.12 Random Sampling. Random Numbers, [783] 18.13 Estimation of Parameters, [785] 18.14 Confidence Intervals, [790] 18.15 Testing of Hypotheses, Decisions, [799] 18.16 Quality Control, [808] 18.17 Acceptance Sampling, [811] 18.18 Goodness of Fit. x2-Test, [817] 18.19 Nonparametric Tests, [819] 18.20 Pairs of Measurements. Fitting Straight Lines, [822] 18.21 Statistical Tables, [828] Appendix 1 References [841] Appendix 2 Answers to Odd-Numbered Problems [845] Index [882]
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Bibliografía: p. 841-844.
CONTENTS --
Introduction Review of Some Topics From Algebra and Calculus [1] --
0.1 Elementary Functions, [1] --
0.2 Partial Derivatives, [8] --
0.3 Second- and Third-Order Determinants, [10] --
0.4 Complex Numbers, [18] --
0.5 Polar Form of Complex Numbers, [22] --
0.6 Some General Remarks About Numerical Computations, [24] --
0.7 Solution of Equations, [26] --
0.8 Approximate Integration, [31] --
Chapter 1 Ordinary Differential Equations of the First Order [39] --
1.1 Basic Concepts and Ideas, [39] --
1.2 Geometrical Considerations. Isoclines, [46] --
1.3 Separable Equations, [49] --
1.4 Equations Reducible to Separable Form, [57] --
1.5 Exact Differential Equations, [59] --
1.6 Integrating Factors, [61] --
1.7 Linear First-Order Differential Equations, [63] --
1.8 Variation of Parameters, [67] --
1.9 Electric Circuits, [69] --
1.10 Families of Curves. Orthogonal Trajectories, [74] --
1.11 Picard’s Iteration Method, [79] --
1.12 Existence and Uniqueness of Solutions, [82] --
1.13 Numerical Methods for Differential Equations of the First Order, [86] --
Chapter 2 Ordinary Linear Differential Equations [93] --
2.1 Homogeneous Linear Equations of the Second Order, [94] --
2.2 Homogeneous Second-Order Equations with Constant Coefficients, [97] --
2.3 General Solution. Fundamental System, [99] --
2.4 Complex Roots of the Characteristic Equation. Initial Value Problem, [102] --
2.5 Double Root of the Characteristic Equation, [106] --
2.6 Free Oscillations, [108] --
2.7 Cauchy Equation, [116] --
2.8 Existence and Uniqueness of Solutions, [117] --
2.9 Homogeneous Linear Equations of Arbitrary Order, [123] --
2.10 Homogeneous Linear Equations of Arbitrary Order with Constant Coefficients, [126] --
2.11 Nonhomogeneous Linear Equations, [128] --
2.12 A Method for Solving Nonhomogeneous Linear Equations, [130] --
2.13 Forced Oscillations. Resonance, [134] --
2.14 Electric Circuits, [140] --
2.15 Complex Method for Obtaining Particular Solutions, [144] --
2.16 General Method for Solving Nonhomogeneous Equations, [147] --
2.17 Numerical Methods for Second-Order Differential Equations, [149] --
Chapter 3 Power Series Solutions of Differential Equations [155] --
3.1 The Power Series Method, [155] --
3.‘2 Theoretical Basis of the Power Series Method, [159] --
3.3 Legendre’s Equation. Legendre Polynomials, [164] --
3.4 Extended Power Series Method. Indicial Equation, [168] --
3.5 Bessel’s Equation. Bessel Functions of the First Kind, [179] --
3.6 Further Properties of Bessel Functions of the First Kind, [185] --
3.7 Bessel Functions of the Second Kind, [187] --
Chapter 4 Laplace Transformation [192] --
4.1 Laplace Transform. Inverse Transform. Linearity, [192] --
4.2 Laplace Transforms of Derivatives and Integrals, [198] --
4.3 Transformation of Ordinary Differential Equations, [201] --
4.4 Partial Fractions, [204] --
4.5 Examples and Applications, [209] --
4.6 Differentiation and Integration of Transforms, [213] --
4.7 Unit Step Function, [216] --
4.8 Shifting on the t-axis, [219] --
4.9 Periodic Functions, [223] --
4.10 Table 17. Some Laplace Transforms, [231] --
Chapter 5 Vector Analysis [235] --
5.1 Scalars and Vectors, [235] --
5.2 Components of a Vector, [237] --
5.3 Vector Addition. Multiplication by Scalars, [239] --
5.4 Scalar Product, [242] --
5.5 Vector Product, [247] --
5.6 Vector Products in Terms of Components, [248] --
5.7 Scalar Triple Product. Linear Dependence of Vectors, [254] --
5.8 Other Repeated Products, [258] --
5.9 Scalar Fields and Vector Fields, [260] --
5.10 Vector Calculus, [263] --
5.11 Curves, [265] --
5.12 Arc Length, [268] --
5.13 Tangent. Curvature and Torsion, [270] --
5.14 Velocity and Acceleration, [274] --
5.15 Chain Rule and Mean Value Theorem for Functions of Several Variables, [278] --
5.16 Directional Derivative. Gradient of a Scalar Field, [282] --
5.17 Transformation of Coordinate Systems and Vector Components, [288] --
5.18 Divergence of a Vector Field, [292] --
5.19 Curl of a Vector Field, [296] --
Chapter 6 Line and Surface Integrals. Integral Theorems [299] --
6.1 Line Integral, [299] --
6.2 Evaluation of Line Integrals, [302] --
6.3 Double Integrals, [306] --
6.4 Transformation of Double Integrals into Line Integrals, [313] --
6.5 Surfaces, [318] --
6.6 Tangent Plane. First Fundamental Form, Area, [321] --
6.7 Surface Integrals, [327] --
6.8 Triple Integrals. Divergence Theorem of Gauss, [332] --
6.9 Consequences and Applications of the Divergence Theorem, [336] --
6.10 Stokes’s Theorem, [342] --
6.11 Consequences and Applications of Stokes’s Theorem, [346] --
6.12 Line Integrals Independent of Path, [347] --
Chapter 7 Matrices and Determinants. Systems of Linear Equations [356] --
7.1 Basic Concepts. Addition of Matrices, [356] --
7.2 Matrix Multiplication, [361] --
7.3 Determinants, [368] --
7.4 Submatrices. Rank, [380] --
7.5 Systems of n Linear Equations in n Unknowns. Cramer’s Rule, [382] --
7.6 Arbitrary Homogeneous Systems of Linear Equations, [386] --
7.7 Arbitrary Nonhomogeneous Systems of Linear Equations, [392] --
7.8 Further Properties of Systems of Linear Equations, [395] --
7.9 Gauss’s Elimination Method, [398] --
7.10 The Inverse of a Matrix, [401] --
7.11 Eigenvalues. Eigenvectors, [406] --
7.12 Bilinear, Quadratic, Hermitian, and Skew-Hermitian Forms, [413] --
7.13 Eigenvalues of Hermitian, Skew-Hermitian, and Unitary Matrices, [417] --
7.14 Bounds for Eigenvalues, [422] --
7.15 Determination of Eigenvalues by Iteration, [426] --
Chapter 8 Fourier Series and Integrals [431] --
8.1 Periodic Functions. Trigonometric Series, [431] --
8.2 Fourier Series. Euler’s Formulas, [434] --
8.3 Even and Odd Functions, [440] --
8.4 Functions Having Arbitrary Period, [444] --
8.5 Half-Range Expansions, [447] --
8.6 Determination of Fourier Coefficients Without Integration, [451] --
8.7 Forced Oscillations, [455] --
8.8 Numerical Methods for Determining Fourier Coefficients. Square Error, [458] --
8.9 Instrumental Methods for Determining Fourier Coefficients, [464] --
8.10 The Fourier Integral, [465] --
8.11 Orthogonal Functions, [473] --
8.12 Sturm-Liouville Problem, [476] --
8.13 Orthogonality of Bessel Functions, [483] --
Chapter 9 Partial Differential Equations [486] --
9.1 Basic Concepts, [486] --
9.2 Vibrating String. One-Dimensional Wave Equation, [488] --
9.3 Separation of Variables (Product Method), [490] --
9.4 D’Alembert’s Solution of the Wave Equation, [499] --
9.5 One-Dimensional Heat Flow, [501] --
9.6 Heat Flow in an Infinite Bar, [506] --
9.7 Vibrating Membrane. Two-Dimensional Wave Equation, [510] --
9.8 Rectangular Membrane, [512] --
9.9 Laplacian in Polar Coordinates, [519] --
9.10 Circular Membrane. Bessel’s Equation, [521] --
9.11 Laplace’s Equation. Potential, [526] --
9.12 Laplace’s Equation in Spherical Coordinates. Legendre’s Equation, [529] --
Chapter 10 Complex Analytic Functions [534] --
10.1 Complex Numbers. Triangle Inequality, [535] --
10.2 Limit. Derivative. Analytic Function, [539] --
10.3 Cauchy*Riemann Equations. Laplace’s Equation, [543] --
10.4 Rational Functions. Root, [547] --
10.5 Exponential Function, [550] --
10.6 Trigonometric and Hyperbolic Functions, [553] --
10.7 Logarithm. General Power, [556] --
Chapter 11 Conformal Mapping [560] --
11.1 Mapping, [560] --
11.2 Conformal Mapping, [564] --
11.3 Linear Transformations, [568] --
11.4 Special Linear Transformations, [572] --
11.5 Mapping by Other Elementary Functions, [577] --
11.6 Riemann Surfaces, [584] --
Chapter 12 Complex Integrals [588] --
12.1 Line Integral in the Complex Plane, [588] --
12.2 Basic Properties of the Complex Line Integral, [594] --
12.3 Cauchy’s Integral Theorem, [595] --
12.4 Evaluation of Line Integrals by Indefinite Integration, [602] --
12.5 Cauchy’s Integral Formula, [605] --
12.6 The Derivatives of an Analytic Function, [607] --
Chapter 13 Sequences and Series [611] --
13.1 Sequences, [611] --
13.2 Series, [618] --
13.3 Tests for Convergence and Divergence of Series, [623] --
13.4 Operations on Series, [629] --
13.5 Power Series, [633] --
13.6 Functions Represented by Power Series, [640] --
Chapter 14 Taylor and Laurent Series [646] --
14.1 Taylor Series, [646] --
14.2 Taylor Series of Elementary Functions, [650] --
14.3 Practical Methods for Obtaining Power Series, [652] --
14.4 Uniform Convergence, [655] --
14.5 Laurent Series, [663] --
14.6 Behavior of Functions at Infinity, [668] --
Chapter 15 Integration by the Method of Residues [671] --
15.1 Zeros and Singularities, [671] --
15.2 Residues, [675] --
15.3 The Residue Theorem, [678] --
15.4 Evaluation of Real Integrals, [680] --
Chapter 16 Complex Analytic Functions and Potential Theory [689] --
16.1 Electrostatic Fields, [689] --
16.2 Two-Dimensional Fluid How, [693] --
16.3 Special Complex Potentials, [697] --
16.4 General Properties of Harmonic Functions, [702] --
16.5 Poisson’s Integral Formula, [705] --
Chapter 17 Special Functions. Asymptotic Expansions [710] --
VIA Gamma and Beta Functions, [710] --
17.2 Error Function. Fresnel Integrals. Sine and Cosine Integrals, [716] --
17.3 Asymptotic Expansions, [720] --
17.4 Further Properties of Asymptotic Expansions, [725] --
Chapter 18 Probability and Statistics [732] --
18.1 Nature and Purpose of Mathematical Statistics, [732] --
18.2 Tabular and Graphical Representation of Samples, [734] --
18.3 Sample Mean and Sample Variance, [740] --
18.4 Random Experiments, Outcomes, Events, [744] --
18.5 Probability, [748] --
18.6 Permutations and Combinations, [752] --
18.7 Random Variables. Discrete and Continuous Distributions, [756] --
18.8 Mean and Variance of a Distribution, [762] --
18.9 Binomial, Poisson, and Hypergeometric Distributions, [766] --
18.10 Normal Distribution, [770] --
18.11 Distributions of Several Random Variables, [777] --
18.12 Random Sampling. Random Numbers, [783] --
18.13 Estimation of Parameters, [785] --
18.14 Confidence Intervals, [790] --
18.15 Testing of Hypotheses, Decisions, [799] --
18.16 Quality Control, [808] --
18.17 Acceptance Sampling, [811] --
18.18 Goodness of Fit. x2-Test, [817] --
18.19 Nonparametric Tests, [819] --
18.20 Pairs of Measurements. Fitting Straight Lines, [822] --
18.21 Statistical Tables, [828] --
Appendix 1 References [841] --
Appendix 2 Answers to Odd-Numbered Problems [845] --
Index [882] --
MR, 35 #7622
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