Mathmatical Methods in the Physical Sciences
6 Vector Analysis
7 Fourier Series and Transforms
8 Ordinary Differential Equations
9 Calculus of Variations
10 Tensor Analysis
1 INTRODUCTION
2 CARTESIAN TENSORS
![Graphics:3 PlotLabel /. Options[{第3章第7节强调的是主动变换（坐标轴固定，矢量旋转），现在要考虑被动变换（矢量固定，坐标轴旋转）, , GraphicsBox[TagBox[RasterBox[RawArray[System`Convert`CommonDump`ConvertText[Byte, System`Convert`HTMLDump`htmlsave, HTMLEntities -> {HTMLBasic}, AltMathOutput -> 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, 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{UseUnicodePlane1Characters -> False, IncludeMarkupAnnotations -> False, Entities -> MathML}], Interleaving -> True], Selectable -> False], BaseStyle -> System`Convert`CommonDump`ConvertText[ImageGraphics, System`Convert`HTMLDump`htmlsave, HTMLEntities -> {HTMLBasic}, AltMathOutput -> 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 -> {201.5, Automatic}, ImageSizeRaw -> {206, 236}, PlotRange -> {{0, 206}, {0, 236}}], , GraphicsBox[TagBox[RasterBox[RawArray[System`Convert`CommonDump`ConvertText[Byte, System`Convert`HTMLDump`htmlsave, HTMLEntities -> {HTMLBasic}, AltMathOutput -> 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}], ArrayObject[Byte, <72252>]], {{0, 108}, {223, 0}}, {0, 255}, ColorFunction -> RGBColor], BoxForm`ImageTag[System`Convert`CommonDump`ConvertText[Byte, System`Convert`HTMLDump`htmlsave, HTMLEntities -> {HTMLBasic}, AltMathOutput -> 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}], ColorSpace -> System`Convert`CommonDump`ConvertText[RGB, System`Convert`HTMLDump`htmlsave, HTMLEntities -> {HTMLBasic}, AltMathOutput -> 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}], Interleaving -> True], Selectable -> False], BaseStyle -> System`Convert`CommonDump`ConvertText[ImageGraphics, System`Convert`HTMLDump`htmlsave, HTMLEntities -> {HTMLBasic}, AltMathOutput -> 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 -> {201.75, Automatic}, ImageSizeRaw -> {223, 108}, PlotRange -> {{0, 223}, {0, 108}}], , 表中是 (x, y, z) 轴与 (x ', y ', z ') 轴之夹角的余弦, , 笛卡尔矢量的定义, , 笛卡尔矢量V在每个矩形坐标系下由三个数（分量）的集合组成, , V ' = ∑ a V ，矩形形式是V ' = A V, , V = ∑ a V ' ，矩阵形式是V = A V ', , a是旋转矩阵A中的方向余弦, , A = (l m n ) , , 笛卡尔张量的定义, , 0阶张量有1个分量，在轴旋转下不变，称为不变量或者标量。比如矢量的长度，或者两矢量的点积, , 1阶张量就是矢量, , 2阶张量（在三维中）在每个矩形坐标系下有9个分量, , 称一个系中的分量为T , 那么一个旋转后的坐标系中的分量是, , (2.14) T ' = ∑ ∑ a a T , k, l = 1, 2, 3, , 直积（direct product），一个二阶张量的例子, , 令U和V是矢量，（在各个坐标系中）从它们的分量构造一个数组, , ( ), , 这9个量是一个二阶张量的分量，表示为U V，这不是点积也不是叉积，是叫做U和V的直积（或者外积，或者张量积）, , U和V在旋转坐标系下的分量是, , U ' = ∑ a U , V ' = ∑ a V , , 那么, , T ' = U ' V ' = ∑ a U ∑ a V = ∑ a a U V = ∑ ∑ a a T , , 方程 (2.14) 立即可以推广, 比如，4 阶笛卡尔张量在每个矩形坐标系下是3 （81） 个分量的集合T ，变换方程是, , T ' = ∑ a a a a T , , 矢量和一个3阶张量的直积就是一个4阶张量，一般的，n阶张量和m阶张量的直积就是m + n阶张量}]j = 1 ij j i j = 1 ji;i = 1 j = 1 ki lj ij j = 1 lj j kl k l i = 1 ki i j = 1 lj j i, j = 1 ki lj i j i = 1 j = 1 ki lj ij i, j, k, l = 1 αi βj γk δl ijkl 3](HTMLFiles/MMPS10_3.gif)
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3 TENSOR NOTATION AND OPERATIONS
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4 INERTIA TENSOR
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5 KRONECKER DELTA AND LEVI-CIVITA SYMBOL
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6 PSEUDOVECTORS AND PSEUDOTENSORS
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7 MORE ABOUT APPLICATIONS
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8 CURVILINEAR COORDINATES
![Graphics: 2 PlotLabel /. Options[{ds = dx + dy + dz (矩形坐标), , ds = dr + r dθ + dz (柱坐标), , ds的表达式称为线元，线元比计算弧长有更多的重要性, , 寻找ds的系统的方法，从, , x = r cosθ, , y = r sinθ, , z = z, , 得到, , dx = cosθ dr - r sinθ dθ, , dy = sinθ dr + r cosθ dθ, , dz = dz, , 平方再求和, , ds = dx + dy + dz = dr + r dθ + dz , , 交叉乘积 (dr dθ) 消掉了。交叉乘积不是一定会消掉，如果消掉的话，就称坐标系正交, , 几何上，正交的坐标系意味着坐标面互相垂直, , GraphicsBox[TagBox[RasterBox[RawArray[System`Convert`CommonDump`ConvertText[Byte, System`Convert`HTMLDump`htmlsave, HTMLEntities -> {HTMLBasic}, AltMathOutput -> 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}], ArrayObject[Byte, <153999>]], {{0, 213}, {241, 0}}, {0, 255}, ColorFunction -> RGBColor], BoxForm`ImageTag[System`Convert`CommonDump`ConvertText[Byte, System`Convert`HTMLDump`htmlsave, HTMLEntities -> {HTMLBasic}, AltMathOutput -> 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}], ColorSpace -> System`Convert`CommonDump`ConvertText[RGB, System`Convert`HTMLDump`htmlsave, HTMLEntities -> {HTMLBasic}, AltMathOutput -> 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}], Interleaving -> True], Selectable -> False], BaseStyle -> System`Convert`CommonDump`ConvertText[ImageGraphics, System`Convert`HTMLDump`htmlsave, HTMLEntities -> {HTMLBasic}, AltMathOutput -> 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 -> {208.25, Automatic}, ImageSizeRaw -> {241, 213}, PlotRange -> {{0, 241}, {0, 213}}], , 柱坐标的坐标面是r = const . (一组同心圆柱面), θ = const . (一组半平面), z = const . (一组平面), , 坐标面相交的三条曲线称为坐标线或者方向, 也是相交于直角。单位基矢量画成和坐标方向相切， 柱坐标中称为e , e , e ，形成正交三元组。这种坐标面不是平面，坐标线是曲线的坐标系称为曲线坐标系。scale factors , , y z不变，x改变dx，那么ds改变dx, , r z不变，θ改变dθ, 那么ds改变rdθ, 这个r就是scale factor, , 求scale factor的方法之一是计算ds , 如果变换是正交的，那么直接可以把scale factor读出来, , 还可以把ds矢量写成ds = e dr + e r dθ + e dz (8.6) , , 可以把曲线坐标系的单位基矢量e , e , e 写成用i, j, k来表示, , 当我们想要对用曲线坐标的基矢量表达的一个矢量进行微分时有用, , ijk的大小和方向都是固定的， 但是e , e 不是, , ds = idx + jdy + kdz = i (cosθ dr - r sinθ dθ) + j (sinθ dr + r cosθ dθ) + kdz = (i cosθ + j sinθ) dr + (-i r sinθ + j r cosθ) dθ + k dz (8.7), , 比较 (8.7) 和 (8.6) ，得到, , e = i cosθ + j sinθ, , r e = -i r sinθ + j r cosθ, , e = k, , 用称为a 和a 的基矢量是比较方便的，不一定要单位长度, , a = e 是一个单位矢量, , a = -i r sinθ + j r cosθ，长度为r, , e = - a = -i sinθ + j cosθ, , 一般的曲线坐标, 令x , x , x 为考虑的坐标变量，比如柱坐标的r, θ, z, , ds = idx + jdy + kdz = i --------- dx + j --------- dx + k --------- dx = a dx + a dx + a dx = a dx , , a = --------- s = i --------- + j --------- + k ---------, , 定义g = a · a , 可以把ds = ds·ds写成矩阵形式, , ds = (dx dx dx ) (g g g ) (dx ), , g 是对称的，因为两向量的点积跟顺序无关, , 用求和约定写成 ds = g dx dx , g 称为度量张量 metric tensor, , 在正交坐标系中，基矢量（e或a）形成正交三元组， 那么ds和ds 可以用scale factor来写, , ds = e h dx + e h dx + e h dx , , ds = (dx dx dx ) ( 2 ) (dx ), , 正交坐标系的体积元是h h h dx dx dx }] ∂ x n ∂ x n ∂ x n 1 1 2 2 3 3 n n n ∂ x ∂ x ∂ x ∂ x&](HTMLFiles/MMPS10_16.gif)
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9 VECTOR OPERATORS IN ORTHOGONAL CURVILINEAR COORDINATES
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10 NON-CARTESIAN TENSORS
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