Advanced Calculus for Applications

Advanced Calculus for Applications

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The text provides advanced undergraduates with the necessary background in advanced calculus topics, providing the foundation for partial differential equations and analysis. Readers of this text should be well-prepared to study from graduate-level texts and publications of similar level. Ordinary Differential Equations; The Laplace Transform; Numerical Methods for Solving Ordinary Differential Equations; Series Solutions of Differential Equations: Special Functions; Boundary-Value Problems and Characteristic-Function Representations; Vector Analysis; Topics in Higher-Dimensional Calculus; Partial Differential Equations; Solutions of Partial Differential Equations of Mathematical Physics; Functions of a Complex Variable; Applications of Analytic Function Theory For all readers interested in advanced more

Product details

  • Hardback | 733 pages
  • 152.4 x 223.52 x 38.1mm | 907.18g
  • Pearson Education (US)
  • Pearson
  • Englewood Cliffs, NJ, United States
  • English
  • 2nd edition
  • 0130111899
  • 9780130111890
  • 1,192,524

Table of contents

1. Ordinary Differential Equations1.1 Introduction1.2 Linear Dependence1.3 Complete Solutions of Linear Equations1.4 The Linear Differential Equation of First Order1.5 Linear Differential Equations with Constant Coefficients1.6 The Equidimensional Linear Differential Equation1.7 Properties of Linear Operators1.8 Simultaneous Linear Differential Equations1.9 particular Solutions by Variation of Parameters1.10 Reduction of Order1.11 Determination of Constants1.12 Special Solvable Types of Nonlinear Equations 2. The Laplace Transform2.1 An introductory Example2.2 Definition and Existence of Laplace Transforms2.3 Properties of Laplace Transforms2.4 The Inverse Transform2.5 The Convolution2.6 Singularity Functions2.7 Use of Table of Transforms2.8 Applications to Linear Differential Equations with Constant Coefficients2.9 The Gamma Function 3. Numerical Methods for Solving Ordinary Differential Equations3.1 Introduction3.2 Use of Taylor Series3.3 The Adams Method3.4 The Modified Adams Method3.5 The Runge-Kutta Method3.6 Picard's Method3.7 Extrapolation with Differences 4. Series Solutions of Differential Equations: Special Functions4.1 Properties of Power Series4.2 Illustrative Examples4.3 Singular Points of Linear Second-Order Differential Equations4.4 The Method of Frobenius4.5 Treatment of Exceptional Cases4.6 Example of an Exceptional Case4.7 A Particular Class of Equations4.8 Bessel Functions4.9 Properties of Bessel Functions4.10 Differential Equations Satisfied by Bessel Functions4.11 Ber and Bei Functions4.12 Legendre Functions4.13 The Hypergeometric Function4.14 Series Solutions Valid for Large Values of x 5. Boundary-Value Problems and Characteristic-Function Representations5.1 Introduction5.2 The Rotating String5.3 The Rotating Shaft5.4 Buckling of Long Columns Under Axial Loads5.5 The Method of Stodola and Vianello5.6 Orthogonality of Characteristic Functions5.7 Expansion of Arbitrary Functions in Series of Orthogonal Functions5.8 Boundary-Value Problems Involving Nonhomogeneous Differential Equations5.9 Convergence of the Method of Stodola and Vianello5.10 Fourier Sine Series and Cosine Series5.11 Complete Fourier Series5.12 Term-by-Term Differentiation of Fourier Series5.13 Fourier-Bessel Series5.14 Legendre Series5.15 The Fourier Integral 6. Vector Analysis6.1 Elementary Properties of Vectors6.2 The Scalar Product of Two Vectors6.3 The Vector Product of Two Vectors6.4 Multiple Products6.5 Differentiation of Vectors6.6 Geometry of a Space Curve6.7 The Gradient Vector6.8 The Vector Operator V6.9 Differentiation Formulas6.10 Line Integrals6.11 The Potential Function6.12 Surface Integrals6.13 Interpretation of Divergence. The Divergence Theorem6.14 Green's Theorem6.15 Interpretation of Curl. Laplace's Equation6.16 Stokes's Theorem6.17 Orthogonal Curvilinear Coordinates6.18 Special Coordinate Systems6.19 Application to Two-Dimensional Incompressible Fluid Flow6.20 Compressible Ideal Fluid Flow 7. Topics in Higher-Dimensional Calculus7.1 Partial Differentiation. Chain Rules7.2 Implicit Functions. Jacobian Determinants7.3 Functional Dependence7.4 Jacobians and Curvilinear Coordinates. Change of Variables in Integrals7.5 Taylor Series7.6 Maxima and Minima7.7 Constraints and Lagrange Multipliers7.8 Calculus of Variations7.9 Differentiation of Integrals Involving a Parameter7.10 Newton's Iterative Method 8. Partial Differential Equations8.1 Definitions and Examples8.2 The Quasi-Linear Equation of First Order8.3 Special Devices. Initial Conditions8.4 Linear and Quasi-Linear Equations of Second Order8.5 Special Linear Equations of Second Order, with Constant Coefficients8.6 Other Linear Equations8.7 Characteristics of Linear First-Order Equations8.8 Characteristics of Linear Second-Order Equations8.9 Singular Curves on Integral Surfaces8.10 Remarks on Linear Second-Order Initial-Value Problems8.11 The Characteristics of a Particular Quasi-Linear Problem 9. Solutions of Partial Differential Equations of Mathematical Physics9.1 Introduction9.2 Heat Flow9.3 Steady-State Temperature Distribution in a Rectangular Plate9.4 Steady-State Temperature Distribution in a Circular Annulus9.5 Poisson's Integral9.6 Axisymmetrical Temperature Distribution in a Solid Sphere9.7 Temperature Distribution in a Rectangular Parallelepiped9.8 Ideal Fluid Flow about a Sphere9.9 The Wave Equation. Vibration of a Circular Membrane9.10 The Heat-Flow Equation. Heat Flow in a Rod9.11 Duhamel's Superposition Integral9.12 Traveling Waves9.13 The Pulsating Cylinder9.14 Examples of the Use of Fourier Integrals9.15 Laplace Transform Methods9.16 Application of the Laplace Transform to the Telegraph Equations for a Long Line9.17 Nonhomogeneous Conditions. The Method of Variation of Parameters9.18 Formulation of Problems9.19 Supersonic Flow of ldeal Compressible Fluid Past an Obstacle 10. Functions of a Complex Variable10.1 Introduction. The Complex Variable10.2 Elementary Functions of a Complex Variable10.3 Other Elementary Functions10.4 Analytic Functions of a Complex Variable10.5 Line Integrals of Complex Functions10.6 Cauchy's Integral Formula10.7 Taylor Series10.8 Laurent Series10.9 Singularities of Analytic Functions10.10 Singularities at Infinity10.11 Significance of Singularities10.12 Residues10.13 Evaluation of Real Definite Integrals10.14 Theorems on Limiting Contours10.15 Indented Contours10.16 Integrals Involving Branch Points 11. Applications of Analytic Function Theory11.1 Introduction11.2 Inversion of Laplace Transforms11.3 Inversion of Laplace Transforms with Branch Points. The Loop Integral 11.4 Conformal Mapping11.5 Applications to Two-Dimensional Fluid Flow11.6 Basic Flows11.7 Other Applications of Conformal Mapping11.8 The Schwarz-Christoffel Transformation11.9 Green's Functions and the Dirichlet Problem11.10 The Use of Conformal Mapping11.11 Other Two-Dimensional Green's Functions Answers to ProblemsIndexContentsshow more

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