Mathmatical Methods in the Physical Sciences
6 Vector Analysis
7 Fourier Series and Transforms
8 Ordinary Differential Equations
1 INTRODUCTION
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2 SEPARABLE EQUATIONS
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3 LINEAR FIRST - ORDER EQUATIONS
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4 OTHER METHODS FOR FIRST - ORDER EQUATIONS
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5 SECOND - ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS AND ZERO RIGHT - HAND SIDE
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6 SECOND - ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS AND RIGHT - HAND SIDE NOT ZERO
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继续
![Graphics: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 d y dy 2 d Y dY 2 i ω ' t i ω ' t ((ω - (ω ') ) - 2b i ω ') F (ω - (ω ') ) F (2b ω ') F (ω - (ω ') ) + (2b ω ') 1 1 iθ F -i φ F i (ω ' t - φ) F 2 2 2 2 2 2 ∞ i n x PlotLabel /. Options[{右边是几个不同指数的和的情况, , principle of superposition, , 分别求特解然后加起来, , 其实就是和的导数等于导数的和这个事实取了个好听的名字, , 这个原理只对线性方程成立，如果含 (y ') , 就不成立了，因为 (y ' + y ') 不等于 (y ') + (y ') , , 满足叠加原理的算符称为线性算符, , 带阻尼的受迫振动, , transient 瞬态振动 : 余函数, , steady - state solution 稳态解：特解, , ---- + 2b -- + ω y = F sin ω ' t, , ---- + 2b -- + ω Y = F e , , 设特解Y = C e , , C = ----------------------------------------------, , {C} = Sqrt[------------------------------------------------- + -------------------------------------------------] = Sqrt[-------------------------------------------------------] F = Sqrt[----------------------------------------------] F = ---------------------------------------------------- F, , C写成r e 形式, , C = ---------------------------------------------------- e , , GraphicsBox[TagBox[RasterBox[RawArray[System`Convert`CommonDump`ConvertText[Byte, System`Convert`HTMLDump`htmlsave, HTMLEntities -> {HTMLBasic}, AltMathOutput -> PlotLabel, WindowSize -> {2000, Automatic}, MathOutput -> GIF, 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PlotLabel, WindowSize -> {2000, Automatic}, MathOutput -> GIF, ConvertClosed -> False, ConvertReverseClosed -> False, ConvertLinkedNotebooks -> False, CharacterEncoding -> Automatic, ConversionStyleEnvironment -> None, ConversionRules -> Automatic, HeadAttributes -> {}, HeadElements -> {}, CSS -> Automatic, ConvertLinkedNotebooks -> False, MathOutput -> GIF, GraphicsOutput -> GIF, Graphics3DOutput -> Automatic, ManipulateOutput -> CDF, ConvertClosed -> True, ConvertReverseClosed -> False, FullDocument -> True, AltMathOutput -> FileName, TableOutput -> {TextForm, Automatic}, AnimationOutput -> Automatic, FilesDirectory -> HTMLFiles, LinksDirectory -> HTMLLinks, HTMLEntities -> {HTML}, AllowBlockMathML -> False, MathMLOptions -> {UseUnicodePlane1Characters -> False, IncludeMarkupAnnotations -> False, Entities -> MathML}], ImageSize -> {211.75, Automatic}, ImageSizeRaw -> {215, 126}, PlotRange -> {{0, 215}, {0, 126}}], , Y = ---------------------------------------------------- e , , 取Y 的虚部, , y = ---------------------------------------------------- sin (ω ' t - φ), , 这就是稳态解，比如电灯开灯之后，随着t增加，余函数（瞬态）变得可忽略，迅速趋于0，稳态解成为全部的解, , 共振：受迫力和y的解是不同相的，因为有相位角φ, 它们的最大值不是同时出现, , 对于一个给定的频率ω ', 频率ω ' = ω的时候，y有最大的幅度，这种情形成为共振, , ω是自然的（无阻的）频率, , 给定频率ω, ω ' 为多少的时候y有最大的幅度？求分母极值， (ω ') = ω - 2 b ，这个ω ' 不等于自然无阻频率， 也不等于自然有阻频率β = ω - b , 但是这个ω ' 是共振频率, , 力学中对于这个ω ' 特别感兴趣，例如对于桥梁，就希望避免频率为ω ' 的周期性的力，因为这种力会产生大的振动, , 收音机调频率的时候，就是给定了ω ' ，然后调整电路使ω = ω ' , , 右边有可能不是正弦余弦函数，而是一个别的周期函数，这时就把右边展开成傅里叶级数的指数形式 ∑ c e ，对每一项求解，然后用叠加原理加起来}] 1 2 1 2 2 dt 2 dt p 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 p 2 2 2 2 2 p p 2 2 2 2 2 n = -∞ n dt dt (ω - (ω ') ) + 4 b (ω ') ((ω - (ω ') ) + 4 b (ω ') ) ((ω - (ω ') ) + 4 b (ω ') ) ((ω - (ω ') ) + 4 b (ω ') ) (ω - (ω ') ) + 4 b (ω ') Sqrt[(ω - (ω ') ) + 4 b (ω ') ] Sqrt[(ω - (ω ') ) + 4 b (ω ') ] Sqrt[(ω - (ω ') ) + 4 b (ω ') ] Sqrt[(ω - (ω ') ) + 4 b (ω ') ]](HTMLFiles/MMPS6-10_8.gif)
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7 OTHER SECOND - ORDER EQUATIONS
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8 THE LAPLACE TRANSFORM
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9 SOLUTION OF DIFFERENTIAL EQUATIONS BY LAPLACE TRANSFORMS
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10 CONVOLUTION
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11 THE DIRAC DELTA FUNCTION
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12 A BRIEF INTRODUCTION TO GREEN FUNCTIONS
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13 MISCELLANEOUS PROBLEMS
9 Calculus of Variations
1 INTRODUCTION
Problems
2 THE EULER EQUATION
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3 USING THE EULER EQUATION
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4 THE BRACHISTOCHRONE PROBLEM; CYCLOIDS
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5 SEVERAL DEPENDENT VARIABLES; LAGRANGE' S EQUATIONS
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6 ISOPERIMETRIC PROBLEMS
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7 VARIATIONAL NOTATION
