下载文档原格式
格林定律：从1806年开始，格林兄弟就致力于民间童话和传说的搜集、整理和研究工作，出版了《儿童和家庭童话集》（两卷集）和《德国传说集》（两卷）。
雅科布还出版了《德国神话》，威廉出版了《论德国古代民歌》和《德国英雄传说》。
1806～1826年间雅科布同时还研究语言学，编写了4卷巨著《德语语法》，是一部历史语法，后人称为日耳曼格语言的基本教程。
在《德语语法》1822年的修订版中，他提出了印欧诸语言语音演变的规则，后人称之为格林定律。
他指出，在印欧语系中日耳曼语族历史上，辅音分组演变，在英语和低地德语中变了一次，后来在高地德语中又再变一次。
事实上，格林定律只是大体上正确，后来由K.A.维尔纳加以补充。
1838年底格林兄弟开始编写《德语词典》，1854～1862 年共出版第一至三卷。
这项浩大的工程兄弟俩生前未能完成，后来德国语言学家继续这项工作，至1961年才全部完成。
印欧语系含大部分欧洲语言和印度次大陆语言在内的约150种语言。
英国语言学家Sir William Jones1786年指出梵语与希腊语和拉丁语可能来自同一个原始语，它们具有亲缘关系。
1822年，Jacob Grimm发现了日耳曼语言中所发生的一系列的有规则的辅音变化。
这些辅音的有规则变化后被称为格林定律：a. 浊爆破音变为清爆破音: bàpb. 清爆破音变为摩擦音: pàfc. 浊送气音变为浊不送气音:bhàb通过比较法重建了被称为原始印欧语系的具有同一来源的语法，包括欧洲语言和印度次大陆的语言的许多亚语系都是以该原始语演化发展来的。
Grimm's law (also known as the First Germanic Sound Shift or the Rask's-Grimm's rule), named for Jacob Grimm, is a set of statements describing the inherited Proto-Indo-European (PIE) stops as they developed in Proto-Germanic (PGmc, the common ancestor of the Germanic branch of the Indo-European family) in the 1st millennium BC. It establishes a set of regular correspondences between early Germanic stops and fricatives and the stop consonants of certain other centum Indo-European languages (Grimm used mostly Latin and Greek for illustration). As it is presently formulated, Grimm's Law consists of three parts, which must be thought of as three consecutive phases in the sense of a chain shift:[1]Proto-Indo-European voiceless stops change into voiceless fricatives.Proto-Indo-European voiced stops become voiceless stops.Proto-Indo-European voiced aspirated stops become voiced fricatives; ultimately, in most Germanic languages these voiced fricatives become voiced stops.The chain shift can be abstractly represented as:bʰ→ b → p → fdʰ→ d → t → θgʰ→ g → k → xgʷʰ→ gʷ→ kʷ→ xʷHere each sound moves one position to the right to take on its new sound value.The voiced aspirated stops may have first become voiced fricatives before hardening to the voiced unaspirated stops "b", "d", and "g" under certain conditions; however, some linguists dispute this. See Proto-Germanic phonology.Grimm's law was the first non-trivial systematic sound change to be discovered in linguistics; its formulation was a turning point in the development of linguistics, enabling the introduction of a rigorous methodology to historical linguistic research. The "law" was discovered by Friedrich von Schlegel in 1806 and Rasmus Christian Rask in 1818. It was elaborated (i.e. extended to include standard German) in 1822 by Jacob Grimm, the elder of the Brothers Grimm, in his book Deutsche Grammatik.Further changes following Grimm's Law, as well as sound changes in other Indo-European languages, can sometimes obscureNote: Some linguists dispute the origin of the word "wife". Calvert Watkins has assumed the root word is Proto-Indo-European *gʷʰíbʰ-. [1]Note: Proto-Germanic *gw from Proto-Indo-Eropean *gʷʰhas undergone further changes of various sorts. After *n it was preserved as *gw, but later changed to *g except in Gothic. Elsewhere, it became either *w or *g during late Proto-Germanic. This is strikingly regular. Each phase involves one single change which applies equally to the labials (p, b, bʰ, f) and their equivalent dentals (t, d, dʰ, þ), velars (k, g, gʰ, h) and rounded velars (kʷ, gʷ, gʷʰ, hw). The first phase left the phoneme repertoire of the language without voiceless stops, the second phase filled this gap but created a new one, and so on until the chain had run its course.Note: Icelandic hv has actually reverted Grimm's Law in the last few generations, and is now pronounced [kʰv] or [kʰf]. Cf. also nynorsk kv-/k-.Some linguists dispute the origin of the word "scold", but Julius Pokorny among others proposed *skwetlo as the assumed root. Dutch has *k → *h (ch) even after *s, though this is a separate development.Furthermore, the voiceless stop *t also did not become a fricative if preceded by *p, *k, or *kʷ (themselves voiceless stops). The voiceless stop it was preceded by did fricativize, however. This is sometimes treated separately under the heading[t:] before pre-aspirating. Thus, the [h] of the modern Icelandic form is not a direct descendant of ancient /h/.[2]The same ancestry holds for the /tt/ of Icelandic átta as well.[3]The most recalcitrant set of apparent exceptions to Grimm's Law, which defied linguists for a few decades, eventually received explanation from the Danish linguist Karl Verner (see the article on Verner's law for details).Correspondences to PIEThe Germanic "sound laws", combined with regular changes reconstructed for other Indo-European languages, allow one to define the expected sound correspondences between different branches of the family. For example, Germanic (word-initial) *b- corresponds regularly to Latin *f-, Greek pʰ-, Sanskrit bʰ-, Slavic, Baltic or Celtic b-, etc., while Germanic *f- corresponds to Latin, Greek, Sanskrit, Slavic and Baltic p- and to zero (no initial consonant) in Celtic. The former set goes back to PIE *bʰ- (faithfully reflected in Sanskrit and modified in various ways elsewhere), and the latter set to PIE *p- (shifted in Germanic, lost in Celtic, but preserved in the other groups mentioned here).GRIMM'S LAW & VERNER'S LAWMajor Changes from I-E to GermanicLarge number of words without known IE cognates. Some NE forms include broad, drink, drive, fowl, hold, meat, rain, and wife.Only two tenses: present and preterit (past)Preterit tense formed with dental suffix (d or t)"Strong" verbs change their tense by internal changese.g., rise-rose, sing-sang"Weak" verbs change tense by adding the dental suffix (-ed)Weak & strong declensions of adjectiveslost in Modern EnglishRegular stress of the first syllablecompare Latin Viri' - viro'rum or ha'beo - habe'musI-E vowels underwent Germanic modificationI-E stops underwent the "First Sound Shift" explained by Grimm's LawGrimm's LawJacob Grimm, 1827German linguist attempted to explain why many Germanic words differed so systematically from their I-E cognates. His formulation (later refined) is called Grimm's Law or the First Sound Shift. High German underwent a Second Sound Shift, but that won't concern our study of English language history.I-E stops gradually assumed new soundsbh --> b dhh --> d ghh --> g ph --> f th --> (theta) kh --> h bh --> p dh --> t gh --> kVerner's LawKarl Verner, 1875Danish linguist wondered why not every I-E stop changed in the same way. His formulation established that Grimm's Law was consistent and could account for all known cognate evolutionIntermediate step in Stage 1 shift:All voiceless stops changed once:ph --> f th --> theta kh --> h sh --> s zIf the sound was in an initial position or immediately after a stressed verb, it changed no further.Those in other positions changed to voiced spirants (b, d, g)格拉斯曼定律格拉斯曼定律是一项用来描述印欧语语音递变的定律，由德国的格拉斯曼(Hermann Grassmann)提出，以补充格里姆定律的不足。
格林公式及其应用一元函数积分学:)()()('a F b F dx x F ba -=⎰§17-1 格林公式及曲线积分与路径无关的条件１ 格林公式的内容格林公式是高等教学中一个著名的计算公式,它建立了曲线积分与二重积分之间的联系．它的条件,结论叙述如下： １.１ 单连通区域设为一平面或空间区域,对于内任意一条闭曲线,总可以在内连续的收缩成内一点则称为单连通区域,否则称是多连通区域． １.２ 格林公式Ⅰ设是平面有界闭域,是有限条封闭的彼此不相交的可求长曲线是并集,则其中表示边界是正向,若是的一条封闭曲线,则定向如下：当人沿进行时,使区域在它的左边,或在上一点作一右手系标架使指向的外法线方向,则的指向即为的方向． １.３ 格林公式Ⅱ设是平面有界闭域,是有限条封闭的彼此不相交的逐段光滑曲线则为边界曲线的外法线方向．例1:计算椭圆12222=+by a x 所围面积A.解: Γ:常数方程 t a x cos = t b y sin =[]ab dt t a t b t b t a ydx xdy A ππ=-⋅-⋅=-=⎰⎰Γ20)sin (sin cos cos 2121例2:计算⎰Γ+-=22y x ydxxdy I ,其中Γ是(1)使所含区域D 不含原点的分段光滑封闭曲线,沿正向(2) 含原点但不径原点解:22y x y P +-= 22y x xQ += 22222)(y x x y y p x +-=∂∂=∂∂θ (1) 满足Green Th 连续条件 ⎰⎰⎰==+-=ΓDd y x ydxxdy I 0022σ(2) 不满足Green Th 连续条件选取适当小的0>ε,作圆周 :222ε=+y x (使 全部含于Γ所围区域) 记 +Γ围成D, 于是在1D 内, 格林公式成立 ⎰⎰⎰⎰⎰⎰⎰-++ΓΓΓ=-=+==001D d σ 故⎰⎰+-=+-Γ 2222y x ydxxdy y x ydx xdy 法一:右式πθθθθεθεπ2)sin (cos 2sin ,cos 202=+==========⎰d y x 学数方程法二:右式⎰⎰⎰≤+=⋅==-=222221122επσεεy x G d ydx xdy公式二、平面上单边通区域内曲线积分与路径无关的等价条件概念:曲线积分⎰Γ+Qdy Qdx 与路径无关:⎰⎰ΓΓ+=+12Qdy Pdx Qdy Pdx图示 (且公与B A y y ,有关)定理:),(),,(y x Q y x P 和平面单连通域D 上具连续一阶偏导,则如下四条件等价.(1)xQy P ∂∂=∂∂ D y x ∈),( (2)⎰Γ=+0Qdy Pdx D ∈Γ 分段光滑闭曲线 (3)积分⎰Γ+A BQdy Pdx 在D 内与路径Γ无关,公与A,B 位置有关(4)存在单值函数),(y x u u =, D y x ∈),( 使它全微分 Qdy Pdx dy y u dx x u du +=∂∂+∂∂=即P x u =∂∂ Q yu =∂∂ 证明:同证)2()1(⇔, )3()2(⇔ 下证)1()4(⇒, )4()3(⇒, )1()4(⇒ 存在函数),(y x u 使 dy y x Q dx y x P du ),(),(+= 则),(y x P x u =∂∂ ),(y x Q yu=∂∂ 于是 y P y x u ∂∂=∂∂∂2 x Qx y u ∂∂=∂∂∂2 由条件 x y uy x u ∂∂∂=∂∂∂22 (连续) 故xQy P ∂∂=∂∂ )4()3(⇒ 曲线积分⎰Γ+A BQdy Pdx 仅与 ),(00y x A ,),(y x B 有关, 记⎰+=),(),(00),(y x B y x A Qdy Pdx y x u (说明右式是y x ,函数)下证 P x u =∂∂ Q yu =∂∂xy x u y x x u x u x ∆-∆+=∂∂→∆),(),(lim 0xQdyPdx Qdy Pdx y x x y x y x y x x ∆+-+=⎰⎰∆+→∆),(),(),(),(00000limx dxy x P x QdyPdx xx xx y x x y x x ∆=∆+=⎰⎰∆+→∆∆+→∆),(limlim0),(),(0),(),(lim ),(lim 1y x P y P xxy P x x Th 连续中值===∆∆===→→∆ξξξ⎥⎦⎤⎢⎣⎡=∆+=∆∆∆+===→∆→∆≤≤),(),(lim ),(lim 0010y x P y x x P x x y x x P x x θθθ同理,),(y x Q yu=∂∂ 故 Qdy Pdx dy yu dx x u du +=∂∂+∂∂=推出公式: 图示 CB AC AB +=⋂AC:0y y = 10x x x ≤≤ 0=dy CB:1x x = 10y y y ≤≤ 0=dx 曲线积分计算公式dy y x Q dx y x P Qdy Pdx Qdy Pdx y y y x B y x A x x A B),(),(11100121),(),(0⎰⎰⎰⎰+=+∆+Γ原函数计算公式C dy y y Q dx y x P C Qdy Pdx y x u yy y x y x xx Th ++=++===⎰⎰⎰),(),(),(00000),(),(0过程特D ∈)0,0( ⎰⎰++=xy C dy y x Q dx x P y x u 0),()0,(),( 可证 ),(),(),(0011),(),(1100y x u y x u y x u Qdy Pdx Qdy Pdx A By x B y x A B A -==+=+⎰⎰Γ ------曲线积分的N-2公式 例3:计算dy x xydx OA⎰Γ+22 三路径.解: 图示 xy y x P 2),(= 2),(x y x Q =xQx y P ∂∂==∂∂2 11)002(2212102)1,1()0,0(22=+⋅+⋅=+=+⎰⎰⎰⎰Γdy x dx x dy x xydx dy x xydx OA例4:计算dy y x x y dx x y y x I )sin sin 2()cos cos 2(22-++=⎰Γ.Γ是1)1(22=+-y x 的上半圆周.从)0,0(O 到)0,2(A解:xQy P ∂∂=∂∂.I 值与路径无关0=⋅→y OA 0=O x 1=A x ,0=dy则⎰⎰===→242xdx I OA⎰Γ-=-=2I例5:dy x y x x y dx x y y x I )sin sin 2()cos cos 2(221+-++=⎰Γ.Γ:例5.解一:xQy P ∂∂+∂∂:不能用与路径无关的相关公式. Γ非闭 :才能用Green 公式.原始方法(第二类曲线积分) 图示 ⎩⎨⎧=+=t y t x sin 1cos 几乎不可能解二:(设法满足二之一: Γ闭)x y y x y Pcos 2sin 2+-=∂∂,1sin 2cos 2+-=∂∂y x x y x Q设1Γ:(从A 到O 直线段)0,0,1,0====dy x x y O A ,则1Γ+Γ构成闭曲线,顺进针.1Γ+Γ所围闭域D:πθ≤≤0, θcos 20≤≤r 由Green 公式2)(1πσσ-=-=∂∂-∂∂-=⎰⎰⎰⎰⎰Γ+ΓD Dd d y P x Q (即⎰⎰ΓΓ-=+12π)而dy x y x x y dx x y y x )sin sin 2()cos cos 2(221+-++⎰Γ⎰-==0242xdx故⎰⎰ΓΓ-=--==12421ππI .解三:(设法满足二之另一,xQy P ∂∂=∂∂) .cos cos 22x y y x P += 设y x x y Q sin sin 221-= x Q =2 21Q Q Q +=则xQ y P ∂∂=∂∂1dy Q Pdx ⎰Γ+1与路径无关. dy Q dy Q Pdx I ⎰⎰ΓΓ++=2111⎰⎰⋅++=20cos )cos 1(2πtdtt xdx24π-=例6:(得用曲线积分求)dy y xy x dx y xy x )2()2(2222--+-+的原函数),(y x u . 并求⎰)2,2()0,1(.(其中Γ是从A(1,0)到B(2,2)的曲线段)解:222y xy x P -+= 222y xy x Q --= y x xQ y P 22-=∂∂=∂∂ C dy y xy x dx y xy x y x u y x +--+-+=⎰)2()2(),(222),()0,0(2C y xy y x x C dy y xy x dx x yx+--+=+--+=⎰⎰3223202023131)2(31),()2()2()2,2()0,1(222)2,2()0,1(2-==--+-+⎰y x u dy y xy x dx y xy x例7 设L 是任意一条分段光滑的闭曲线，求证220Lxydx x dy +=⎰证明 令22,P xy Q x ==则P Qx y x∂∂==∂∂在全xOy ，这个单连通区域G 内成立．故由格林公式可得2200LDxydx x dy dxdy +=±=⎰⎰⎰ ．(2)当考虑积分L Pdx Qdy +⎰ 时，若L 为平面区域G 内一条简单闭曲线，而区域G 为含有“点洞”M 的复连通区域，函数P 、Q 除点M 外，处处有连续偏导数存在，且满足P Qy x∂∂=∂∂．当闭路L 不包围点M 时，此曲线积分的值为零．当闭路L 包围点M 时，一般说来，此线积分不再为零，积分值为一常数，具体求法如下：只要选择一个适当小的包围点M 的正向闭曲线C 来将点M 扣掉，则曲线积分在以L 和C 围成的复连通区域G 内仍可用格林公式计算，并有结论：LCPdx Qdy Pdx Qdy +=+⎰⎰其中C 为闭路正向．#综上可知,格林公式可使曲线积分的计算大大简化,因此在场论、流体力学、热力学、电学及微分方程等学科中得到广泛的应用.。
格林公式范文格林公式是数学分析中的一个重要公式，它在向量分析和微积分等领域有着广泛的应用。
格林公式由英国数学家乔治·格林于1828年提出，被视为一种向量微积分定理。
它通过将一些区域的边界和内部连接起来，将曲线积分转换为面积积分，从而简化了许多计算问题。
格林公式的一般形式如下：∫∫(P∂x + Q∂y)dxdy = ∮(Qdx - Pdy),其中∂x和∂y表示区域D内的一个小面元，P和Q是定义在D内的可微函数，∮表示D的边界曲线C的路径积分。
格林公式可以看作是两个不同领域的重要定理之间的关系，即格林定理（Green's Theorem）和斯托克斯定理（Stokes' theorem）。
格林定理是格林公式的一种特殊情况，它用于计算平面区域上的曲线积分。
当Q = dP/dx时，格林公式退化为格林定理：∮(Pdx + Qdy) = ∬(dQ/dx - dP/dy)dxdy,其中∮表示C的路径积分，∬表示D的面积积分。
格林定理在计算曲线积分时非常有用，它可以将曲线积分转化为面积积分，使得计算变得更加简便。
斯托克斯定理是格林公式在三维空间下的推广，用于计算曲面上的曲线积分。
斯托克斯定理表达如下：∮(Pdx + Qdy + Rdz) = ∬(curl F · ndS),其中∮表示曲面S的边界曲线的路径积分，∬表示曲面S的面积积分，F = (P, Q, R)是一个向量场，curl F是该向量场的旋度，n是曲面的单位法向量。
格林公式的推导涉及到高等数学中的一些概念和定理，如多元函数的偏导数、向量场、曲线、曲面等。
它的证明过程可以采用传统的微积分推导方法，如应用泰勒展开和对极限的计算。
此外，格林公式的证明还可以通过使用微分形式和外微分运算的方法进行。
格林公式的应用非常广泛，特别是在物理学和工程学领域。
它可以用于计算流体力学中的流速场、电磁场中的电势和磁场等。
格林公式的应用也为解决各种边值问题提供了便利，如泊松方程、拉普拉斯方程的求解等。
