Advanced Calculus for Applications
Advanced Calculus for Applications
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发表时间：
1962
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通讯作者：
F. B. Hildebrand
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作者：
F. B. Hildebrand
1. 常微分方程1.1简介1.2线性相关性1.3线性方程的完全解1.4一阶线性微分方程1.5常系数线性微分方程1.6等维线性微分方程1.7线性算子的性质1.8联立线性微分方程1.9参数变分的特解1.10降阶1.11常数的确定1.12特殊可解类型非线性方程拉普拉斯变换2.1一个介绍性的例子2.2拉普拉斯变换的定义和存在性2.3拉普拉斯变换的性质2.4逆变换2.5卷积2.6奇异函数2.7变换表的应用2.8常系数线性微分方程的应用2.9函数3常微分方程的数值解法3.1简介3.2泰勒级数的应用3.3亚当斯法3.4修正亚当斯法3.5龙格-库塔法3.6皮卡德法3.7差分外推法4。微分方程的级数解：特殊函数4.1幂级数的性质4.2举例4.3线性二阶微分方程的奇异点4.4 Frobenius方法4.5例外情况的处理4.6例外情况的例子4.7一类特殊方程4.8贝塞尔函数4.9贝塞尔函数的性质4.10贝塞尔函数所满足的微分方程4.11 Ber和Bei函数4.12 Legendre函数4.13超几何函数4.14级数解的有效性对于x 5的大值。边值问题与特征函数表示5.1简介5.2旋转弦5.3转轴5.4轴向载荷下长柱的屈曲5.5 Stodola和Vianello法5.6特征函数的正交性5.7正交函数级数中任意函数的展开5.8涉及非齐次微分方程的边值问题5.9 Stodola和Vianello法的收敛性5.10傅立叶正弦和余弦系列5.11完全傅立叶级数5.12傅立叶级数的逐项微分5.13傅立叶-贝塞尔级数5.14勒让德级数5.15傅立叶积分向量分析6.1向量的基本性质6.2两个向量的标量积6.3两个向量的向量积6.4倍积6.5向量的微分6.6空间曲线的几何6.7梯度向量6.8向量算子V 6.9微分公式6.10线积分6.11势函数6.12曲面积分6.13散度的解释散度定理6.14格林定理6.15旋度的解释拉普拉斯方程6.16斯托克斯定理6.17正交曲线坐标6.18特殊坐标系6.19二维不可压缩流体流动的应用6.20可压缩理想流体流动高维微积分主题7.1偏微分。链式规则7.2隐式函数。雅可比行列式7.3泛函相关性7.4雅可比矩阵与曲线坐标。积分中的变量变换7.5泰勒级数7.6极大极小7.7约束与拉格朗日乘子7.8变分法7.9含参数积分的微分7.10牛顿迭代法偏微分方程8.1定义与实例8.2一阶拟线性方程8.3特殊器件初始条件8.4二阶线性与拟线性方程8.5特殊二阶常系数线性方程8.6其他线性方程8.7线性一阶方程的特征8.8线性二阶方程的特征8.9积分曲面上的奇异曲线8.10线性二阶初值问题的注意事项8.11一类特殊拟线性问题的特征数学物理偏微分方程解9.1绪论9.2热流9.3矩形板内稳态温度分布9.4环内稳态温度分布9.5泊松积分9.6实心球内轴对称温度分布9.7矩形平行六面体内温度分布9.8理想流体绕球流动9.9波动方程圆膜的振动9.10热流方程。棒中的热流9.11 Duhamel的叠加积分9.12行波9.13脉动圆柱9.14傅立叶积分的应用实例9.15拉普拉斯变换方法9.16拉普拉斯变换在长线电报方程中的应用9.17非齐次条件参数变分法9.18问题的表述9.19理想可压缩流体的超音速过障流动10。复变量函数10.1简介。复变量10.2复变量的初等函数10.3其他初等函数10.4复变量的解析函数10.5复函数的线积分10.6柯西积分公式10.7泰勒级数10.8洛朗级数10.9解析函数的奇点10.10无穷远处的奇点10.11奇点的意义10.12残数10.13实定积分的求值10.14极限等值线的定理10.15缩进等值线10.16包含分支点的积分解析函数理论的应用11.1绪论11.2拉普拉斯变换的反演11.3带分支点的拉普拉斯变换的反演环积分11.4保角映射11.5在二维流体流动中的应用11.6基本流动11.7保角映射的其他应用11.8 Schwarz-Christoffel变换11.9 Green函数与Dirichlet问题11.10保角映射的应用11.11其他二维Green函数的问题解答
1. Ordinary Differential Equations 1.1 Introduction 1.2 Linear Dependence 1.3 Complete Solutions of Linear Equations 1.4 The Linear Differential Equation of First Order 1.5 Linear Differential Equations with Constant Coefficients 1.6 The Equidimensional Linear Differential Equation 1.7 Properties of Linear Operators 1.8 Simultaneous Linear Differential Equations 1.9 particular Solutions by Variation of Parameters 1.10 Reduction of Order 1.11 Determination of Constants 1.12 Special Solvable Types of Nonlinear Equations 2. The Laplace Transform 2.1 An introductory Example 2.2 Definition and Existence of Laplace Transforms 2.3 Properties of Laplace Transforms 2.4 The Inverse Transform 2.5 The Convolution 2.6 Singularity Functions 2.7 Use of Table of Transforms 2.8 Applications to Linear Differential Equations with Constant Coefficients 2.9 The Gamma Function 3. Numerical Methods for Solving Ordinary Differential Equations 3.1 Introduction 3.2 Use of Taylor Series 3.3 The Adams Method 3.4 The Modified Adams Method 3.5 The Runge-Kutta Method 3.6 Picard's Method 3.7 Extrapolation with Differences 4. Series Solutions of Differential Equations: Special Functions 4.1 Properties of Power Series 4.2 Illustrative Examples 4.3 Singular Points of Linear Second-Order Differential Equations 4.4 The Method of Frobenius 4.5 Treatment of Exceptional Cases 4.6 Example of an Exceptional Case 4.7 A Particular Class of Equations 4.8 Bessel Functions 4.9 Properties of Bessel Functions 4.10 Differential Equations Satisfied by Bessel Functions 4.11 Ber and Bei Functions 4.12 Legendre Functions 4.13 The Hypergeometric Function 4.14 Series Solutions Valid for Large Values of x 5. Boundary-Value Problems and Characteristic-Function Representations 5.1 Introduction 5.2 The Rotating String 5.3 The Rotating Shaft 5.4 Buckling of Long Columns Under Axial Loads 5.5 The Method of Stodola and Vianello 5.6 Orthogonality of Characteristic Functions 5.7 Expansion of Arbitrary Functions in Series of Orthogonal Functions 5.8 Boundary-Value Problems Involving Nonhomogeneous Differential Equations 5.9 Convergence of the Method of Stodola and Vianello 5.10 Fourier Sine Series and Cosine Series 5.11 Complete Fourier Series 5.12 Term-by-Term Differentiation of Fourier Series 5.13 Fourier-Bessel Series 5.14 Legendre Series 5.15 The Fourier Integral 6. Vector Analysis 6.1 Elementary Properties of Vectors 6.2 The Scalar Product of Two Vectors 6.3 The Vector Product of Two Vectors 6.4 Multiple Products 6.5 Differentiation of Vectors 6.6 Geometry of a Space Curve 6.7 The Gradient Vector 6.8 The Vector Operator V 6.9 Differentiation Formulas 6.10 Line Integrals 6.11 The Potential Function 6.12 Surface Integrals 6.13 Interpretation of Divergence. The Divergence Theorem 6.14 Green's Theorem 6.15 Interpretation of Curl. Laplace's Equation 6.16 Stokes's Theorem 6.17 Orthogonal Curvilinear Coordinates 6.18 Special Coordinate Systems 6.19 Application to Two-Dimensional Incompressible Fluid Flow 6.20 Compressible Ideal Fluid Flow 7. Topics in Higher-Dimensional Calculus 7.1 Partial Differentiation. Chain Rules 7.2 Implicit Functions. Jacobian Determinants 7.3 Functional Dependence 7.4 Jacobians and Curvilinear Coordinates. Change of Variables in Integrals 7.5 Taylor Series 7.6 Maxima and Minima 7.7 Constraints and Lagrange Multipliers 7.8 Calculus of Variations 7.9 Differentiation of Integrals Involving a Parameter 7.10 Newton's Iterative Method 8. Partial Differential Equations 8.1 Definitions and Examples 8.2 The Quasi-Linear Equation of First Order 8.3 Special Devices. Initial Conditions 8.4 Linear and Quasi-Linear Equations of Second Order 8.5 Special Linear Equations of Second Order, with Constant Coefficients 8.6 Other Linear Equations 8.7 Characteristics of Linear First-Order Equations 8.8 Characteristics of Linear Second-Order Equations 8.9 Singular Curves on Integral Surfaces 8.10 Remarks on Linear Second-Order Initial-Value Problems 8.11 The Characteristics of a Particular Quasi-Linear Problem 9. Solutions of Partial Differential Equations of Mathematical Physics 9.1 Introduction 9.2 Heat Flow 9.3 Steady-State Temperature Distribution in a Rectangular Plate 9.4 Steady-State Temperature Distribution in a Circular Annulus 9.5 Poisson's Integral 9.6 Axisymmetrical Temperature Distribution in a Solid Sphere 9.7 Temperature Distribution in a Rectangular Parallelepiped 9.8 Ideal Fluid Flow about a Sphere 9.9 The Wave Equation. Vibration of a Circular Membrane 9.10 The Heat-Flow Equation. Heat Flow in a Rod 9.11 Duhamel's Superposition Integral 9.12 Traveling Waves 9.13 The Pulsating Cylinder 9.14 Examples of the Use of Fourier Integrals 9.15 Laplace Transform Methods 9.16 Application of the Laplace Transform to the Telegraph Equations for a Long Line 9.17 Nonhomogeneous Conditions. The Method of Variation of Parameters 9.18 Formulation of Problems 9.19 Supersonic Flow of ldeal Compressible Fluid Past an Obstacle 10. Functions of a Complex Variable 10.1 Introduction. The Complex Variable 10.2 Elementary Functions of a Complex Variable 10.3 Other Elementary Functions 10.4 Analytic Functions of a Complex Variable 10.5 Line Integrals of Complex Functions 10.6 Cauchy's Integral Formula 10.7 Taylor Series 10.8 Laurent Series 10.9 Singularities of Analytic Functions 10.10 Singularities at Infinity 10.11 Significance of Singularities 10.12 Residues 10.13 Evaluation of Real Definite Integrals 10.14 Theorems on Limiting Contours 10.15 Indented Contours 10.16 Integrals Involving Branch Points 11. Applications of Analytic Function Theory 11.1 Introduction 11.2 Inversion of Laplace Transforms 11.3 Inversion of Laplace Transforms with Branch Points. The Loop Integral 11.4 Conformal Mapping 11.5 Applications to Two-Dimensional Fluid Flow 11.6 Basic Flows 11.7 Other Applications of Conformal Mapping 11.8 The Schwarz-Christoffel Transformation 11.9 Green's Functions and the Dirichlet Problem 11.10 The Use of Conformal Mapping 11.11 Other Two-Dimensional Green's Functions Answers to Problems Index Contents