查看“︁Faa de Bruno's Theorem”︁的源代码

{{header
 | title       = Faa de Bruno's Theorem
 | author      = E. B. Elliott
 | translator  = 曾炯
 | year        = 1924
 | month       = 4
 | theme       = 數學
 | notes       = 刊於《國立武昌師範大學數理化雜誌》第十二期（十週年紀念號），民國十三年四月，武昌師範大學數理化學會出版。原文取自E. B. Elliott著《An introduction to the algebra of quantics》（1895年初版，1913年二版）。
}}

{{center|{{++|Faa de Bruno's Theorem}}}}

{{center|{{+|曾　炯}}}}

{{center|{{+|<math>lm'-l'm</math>乘冪之行列式}}}}

　因欲直接證明多種函數之爲行列式形者，爲二元形Binary quantic之不變式invariants及共變式covariants，有初簡單定理對於某種行列式極爲有助，即Faa de Bruno's Theorem是也．此定理之最初三類爲：

　　<math>{\begin{vmatrix}l,&m\\l',&m'\end{vmatrix}}=lm' - l'm</math>

　　<math>{\begin{vmatrix}l^2&lm&m^2\\2ll'&lm'+l'm,&2mm'\\l'^2&l'm'&m'^2\end{vmatrix}}=(lm' - l'm)^3</math>

　　<math>{\begin{vmatrix}l^3&l^2m&lm^2&m^3\\3l^2l',&2ll'm+l^2m',&2lmm'+l'm^2&3m^2m'\\3ll'^2,&l'^2m+2ll'm',&lm^2+2l'mm',&3mm'^2\\l'^3&l'^2m'&l'm'^2&m'^2\end{vmatrix}}=(lm' - l'm)^6</math>

而此一般定理General Theorem,乃一行列式,其第一列由下列各元素而成

　　<math>l^r, l^{r-1}m, l^{r-2}m^2, \cdots\cdots\cdots\cdots lm^{r-1},m^r</math>

而其各列可由第一列繼續的in succession與

<math>l'\frac{\partial}{\partial l} + m'\frac{\partial}{\partial m}, \frac 1 {1\cdot 2}\left(l'\frac{\partial}{\partial l} + m' \frac{\partial}{\partial m}\right)^2, \cdots\cdots\cdots\cdots\frac 1 {r!}\left(l'\frac{\partial}{\partial l} + m' \frac{\partial}{\partial m}\right)^r, </math>

演算Operating而得之,此行列式爲<math>lm'-l'm</math>之乘寬即<math>\frac 1 2 r (r+1)</math>''th''冪也.

　此又有顯然易見者,卽可同樣的先書下其最末一列:

　　<math>l'^r, l'^{r-1}m',l'^{r-2}m'^2,\cdots\cdots\cdots\cdots l'm'^{r-1}, m'^r</math>

而繼續的向上upwards 由此列與

　　<math>l \frac{\partial}{\partial l'} + m \frac{\partial}{\partial m'}, \frac 1 {1\cdot 2}\left(l \frac{\partial}{\partial l'} + m \frac{\partial}{\partial m'}\right)^2, \cdots\cdots\cdots\cdots \frac 1 {r!} \left(l\frac{\partial}{\partial l'} + m \frac{\partial}{\partial m'}\right)^r</math>

演算而得其餘各列也.因由Taylor's Theorem,在第<math>(s+1)</math>行之元素向下讀之乃

　　<math>(l+tl')^{r-s}(m+tm')^s</math>

之展開<math>t</math>之各冪Various powers之係數;而同樣向上讀之,乃

　　<math>(\tau l+l')^{r-s}(\tau m+m')^s</math>

之展開<math>\tau</math>之各冪之係數也.

　此兩種形成行列式之方法;謂之第一種寫法及第二種寫法.

　此定理之第一類卽可證明,其第二類之證明:以<math> -m/m'</math>乘第二列,<math>m^2/m'^2</math>乘第三列,加之於第一列,卽可得證明之,第三類之證明亦可由同樣之方法易於求之.此一般定理,乃線偏微分方程式Lagrange解法之定理The theory of Lagrange's solution of linear partial differential equations中,一簡易習題,茲將進而討論之.

　由乘積Products微分之普通之規則,可知微分<math>r</math>次行列式之結果,可書爲<math>r</math>個行列式之和,其每行列式由微分原行列式中一列之元素,而遺留其餘各列元素不動而得之.試思此原行列式爲第一種寫法,以<math>l' \frac{\partial}{\partial l} + m' \frac{\partial}{\partial m}</math>演算之.此結果乃<math>r</math>個行列式之和,而此等列行式皆消滅爲零Vanish因演算任何列之結果,除最末一列外,皆發生爲其下一列following row之數值之倍數a numerical multiple而最末一列演算之結果爲一列零.於是若<math>D</math>表示此原行列式,則得

　　<math>l' \frac{\partial D}{\partial l} + m' \frac{\partial D}{\partial m} = 0,</math>

故由Lagrange定理僅含有<math>l</math>及<math>m</math>於<math>lm'-l'm</math>關係之中.

　再試思<math>D</math>爲第二種寫法,以<math>l \frac{\partial}{\partial l'} + m \frac{\partial}{\partial m'}</math>演算之,同樣得

　　<math>l \frac{\partial D}{\partial l'} + m \frac{\partial D}{\partial m'}=0.</math>

故<math>D</math>僅含有<math>l'</math>及<math>m'</math>於<math>lm'-l'm</math>關係之中.

　是故此行列式<math>D</math>,僅爲<math>lm'-l'm</math>之函數,且爲同次式Homogeneous而必爲<math>lm'-l'm</math>之一單冪single power帶有一可能的數值的因數而成者也.但此數值的因數爲<math>1</math>,例如取<math>l=m'=1</math>, <math>l'=m=0</math>,則<math>lm'-l'm</math>爲<math>1</math>,而<math>D</math>爲一主對角線A principal diagonal之<math>1</math>,及其他各元素零而成.

　由是可知<math>D</math>爲<math>l,m,l',m'</math>之<math>r(r+1)</math>次元Dimension,更由此事實而知<math>D</math>爲<math>lm'-l'm</math>之<math>\frac 1 2 r(t+1)</math>''th''冪矣.

　試證明

　　<math>\begin{vmatrix}\frac{\partial^4 u}{\partial x^4}, & \frac{\partial^4 u}{\partial x^3\partial y}, & \frac{\partial^4 u}{\partial x^2\partial y^2} \\ \frac{\partial^4 u}{\partial x^3\partial y}, & \frac{\partial^4 u}{\partial x^2\partial y^2}, & \frac{\partial^4 u}{\partial x\partial y^3} \\ \frac{\partial^4 u}{\partial x^2\partial y^2}, & \frac{\partial^4 u}{\partial x\partial y^3}, & \frac{\partial^4 u}{\partial y^4}\end{vmatrix}</math>

爲一個二元形<math>u</math>之一共變式，於特別情形,若<math>u</math>爲四次式quartic,則爲一不變式.以爲本定理之標準應用。

　由上述<math>(lm'-l'm)^3</math>卽<math>M^3</math>之行列式，行與行乘此式二次。

　第一次乘法之結果，因

　　<math>\left(l\frac{\partial}{\partial x} + l'\frac{\partial}{\partial y}\right)^2 = \frac{\partial^2}{\partial X^2}, \left(l \frac{\partial}{\partial x} + l'\frac{\partial}{\partial y}\right) \left(m\frac{\partial}{\partial x}+m'\frac{\partial}{\partial y}\right)=\frac{\partial^2}{\partial X\partial Y}</math>

　　<math>\left(m\frac{\partial}{\partial x} + m'\frac{\partial}{\partial y} \right) \equiv \frac{\partial^2}{\partial Y^2}</math>

　　<math>\begin{vmatrix}\frac{\partial^2}{\partial X^2}\cdot\frac{\partial^2 u}{\partial x^2}, & \frac{\partial^2}{\partial X\partial Y}\cdot\frac{\partial^2 u}{\partial x^2}, & \frac{\partial^2}{\partial Y^2}\cdot\frac{\partial^2 u}{\partial x^2} \\
\frac{\partial^2}{\partial X^2}\cdot\frac{\partial^2 u}{\partial x \partial y}, & \frac{\partial^2}{\partial X\partial Y}\cdot\frac{\partial^2 u}{\partial x \partial y}, & \frac{\partial^2}{\partial Y^2}\cdot\frac{\partial^2 u}{\partial x \partial y} \\ \frac{\partial^2}{\partial X^2}\cdot\frac{\partial^2 u}{\partial y^2}, & \frac{\partial^2}{\partial X\partial Y}\cdot\frac{\partial^2 u}{\partial y^2}, & \frac{\partial^2}{\partial Y^2}\cdot\frac{\partial^2 u}{\partial y^2}\end{vmatrix} </math>

　第二次乘法變易其每元素之微分次數,則

　　<math>\begin{vmatrix}\frac{\partial^4 u}{\partial X^4}, &\frac{\partial^4 u}{\partial X^3\partial Y}, &\frac{\partial^4 u}{\partial X^2\partial Y^2} \\ \frac{\partial^4 u}{\partial X^3\partial Y}, &\frac{\partial^4 u}{\partial X^2\partial Y^2}, &\frac{\partial^4 u}{\partial X\partial Y^3} \\ \frac{\partial^4 u}{\partial X^2\partial Y^2}, & \frac{\partial^4 u}{\partial X\partial Y^3} &\frac{\partial^4 u}{\partial Y^4}\end{vmatrix}</math>

　如是上述之事實證明矣

　注意I. 此篇譯自 E. B. Elliott's Algebras of Quantics 中16,17兩節

　注意II. The theory of Lagrange's solution of linear partial differential equations可參考A. R. Forsyth's Differential Euqations之187,189兩節PP.392-394.

　注意III. Faa de Bruno (1825-1888)

{{Translation license|original={{Pd/1923|1937}}|translation={{Pd/1996|1940}}}}