en>Yobot: WP:CHECKWIKI error fixes + general fixes using AWB (7896)

2012-01-03T22:44:15Z

WP:CHECKWIKI error fixes + general fixes using AWB (7896)

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{{other uses2|Leibniz's rule}}

In [[calculus]], the '''general Leibniz rule''',<ref>Olver, Applications of Lie groups to differential equations, page 318</ref> named after [[Gottfried Leibniz]], generalizes the [[product rule]] (which is also known as "[[Leibniz's rule]]".) It states that if ''f'' and ''g'' are ''n''-times differentiable functions, then the ''n''th derivative of the product ''fg'' is given by

:<math>(f \cdot g)^{(n)}=\sum_{k=0}^n {n \choose k} f^{(k)} g^{(n-k)}</math>

where <math>{n \choose k}</math> is the [[binomial coefficient]].

This can be proved by using the product rule and [[mathematical induction]].

With the [[multi-index]] notation the rule states more generally:

:<math>\partial^\alpha (fg) = \sum_{ \{\beta\,:\,\beta \le \alpha \} } {\alpha \choose \beta} (\partial^{\alpha - \beta} f) (\partial^{\beta} g).</math>

This formula can be used to derive a formula that computes the symbol of the composition of differential operators. In fact, let ''P'' and ''Q'' be differential operators (with coefficients that are differentiable sufficiently many times) and <math>R = P \circ Q</math>. Since ''R'' is also a differential operator, the symbol of ''R'' is given by:

:<math>R(x, \xi) = e^{-{\langle x, \xi \rangle}} R (e^{\langle x, \xi \rangle}).</math>

A direct computation now gives:

:<math>R(x, \xi) = \sum_\alpha {1 \over \alpha!} \left({\partial \over \partial \xi}\right)^\alpha P(x, \xi) \left({\partial \over \partial x}\right)^\alpha Q(x, \xi).</math>

This formula is usually known as the Leibniz formula. It is used to define the composition in the space of symbols, thereby inducing the ring structure.

==See also==

*[[Derivation (abstract algebra)]]
*[[Differential algebra]]
*[[Product rule]]
*[[Derivative]]

==Notes==
<references/>

==External links==
*[http://planetmath.org/encyclopedia/GeneralizedLeibnizRule.html Planet Math]

{{DEFAULTSORT:General Leibniz Rule}}
[[Category:Calculus]]
[[Category:Gottfried Leibniz]]

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en>Yobot: WP:CHECKWIKI error fixes + general fixes using AWB (7896)