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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。
格林倒易定理证明格林倒易定理是数学中非常重要的一个定理，它在几何、电磁学和流体力学等领域都有广泛的应用。
该定理由德国数学家格林于1828年首次提出，被称为格林倒易定理或格林公式。
首先，我们来看一下该定理的内容。
格林倒易定理是关于二维平面上的曲线积分和面积积分之间的关系。
简而言之，该定理可以将曲线积分转化为面积积分，或者将面积积分转化为曲线积分。
具体来说，假设有一个平面内的有界区域D，边界为C，且C是一个简单闭合曲线，即不交叉、没有自交的曲线。
若函数P(x, y)和Q(x, y)在D上有一阶连续偏导数，那么有：∮C [P(x, y)dx + Q(x, y)dy] = ∬D [(∂Q/∂x - ∂P/∂y)]dxdy其中，dx和dy分别表示微小位移在x轴和y轴上的分量，∮C表示对曲线C的积分，∬D表示对区域D的积分，∂Q/∂x表示Q对x的偏导数，∂P/∂y表示P对y的偏导数。
这个公式意味着，通过计算曲线C上某个向量场（P, Q）的积分，可以得到区域D内这个向量场的发散，即（∂Q/∂x - ∂P/∂y）。
反之，通过计算区域D内的向量场的发散，可以得到曲线C上该向量场的积分。
格林倒易定理在几何学中的应用非常广泛，可以用于计算曲线的长度、曲线的外法向量、曲率等。
在电磁学中，格林倒易定理可以用于计算电场或磁场的闭合曲线的环流，从而得到电场或磁场的发散。
在流体力学中，格林倒易定理可以用于计算流体速度场的环流和流体密度的发散。
在实际应用中，我们可以利用格林倒易定理简化计算过程。
通过转化曲线积分为面积积分或反之，我们可以将问题从曲线上的积分转化为区域内的双重积分，从而简化计算过程并得到更便于处理的结果。
总之，格林倒易定理作为数学中的重要工具，可以将曲线积分和面积积分之间建立起联系，为几何学、电磁学和流体力学等领域的问题提供了一种统一的解决方法。
在实践中，我们可以通过灵活运用该定理来简化计算，并获得更深入的理解和洞察力。
