Derivation (differential algebra)

In mathematics, a derivation is a function on an algebra that generalizes certain features of the derivative operator. Specifically, given an algebra ${\displaystyle A}$ over a ring or a field${\displaystyle K}$, a ${\displaystyle K}$-derivation is a ${\displaystyle K}$-linear map${\displaystyle D:A\to A}$ that satisfies Leibniz's law:

${\displaystyle D(ab)=aD(b)+D(a)b.}$

More generally, if ${\displaystyle M}$ is an ${\displaystyle A}$-bimodule, a ${\displaystyle K}$-linear map ${\displaystyle D:A\to M}$ that satisfies the Leibniz law is also called a derivation. The collection of all ${\displaystyle K}$-derivations of ${\displaystyle A}$ to itself is denoted by ${\displaystyle \mathrm {Der} _{K}(A)}$. The collection of ${\displaystyle K}$-derivations of ${\displaystyle A}$ into an ${\displaystyle A}$-module ${\displaystyle M}$ is denoted by ${\displaystyle \mathrm {Der} _{K}(A,M)}$.

Derivations occur in many different contexts in diverse areas of mathematics. The partial derivative with respect to a variable is an ${\displaystyle \mathbb {R} }$-derivation on the algebra of real-valued differentiable functions on ${\displaystyle \mathbb {R} ^{n}}$. The Lie derivative with respect to a vector field is an ${\displaystyle \mathbb {R} }$-derivation on the algebra of differentiable functions on a differentiable manifold; more generally it is a derivation on the tensor algebra of a manifold. It follows that the adjoint representation of a Lie algebra is a derivation on that algebra. The Pincherle derivative is an example of a derivation in abstract algebra. If the algebra ${\displaystyle A}$ is noncommutative, then the commutator with respect to an element of the algebra ${\displaystyle A}$ defines a linear endomorphism of ${\displaystyle A}$ to itself, which is a derivation over ${\displaystyle K}$. That is,

${\displaystyle [FG,N]=[F,N]G+F[G,N]\,,}$

where ${\displaystyle [\cdot ,N]}$ is the commutator with respect to ${\displaystyle N}$. An algebra ${\displaystyle A}$ equipped with a distinguished derivation ${\displaystyle d}$ forms a differential algebra, and is itself a significant object of study in areas such as differential Galois theory.

If ${\displaystyle A}$ is a ${\displaystyle K}$-algebra, for ${\displaystyle K}$ a ring, and D: ${\displaystyle A}$ → ${\displaystyle A}$ is a ${\displaystyle K}$-derivation, then

• If ${\displaystyle A}$ has a unit 1, then ${\displaystyle D(1)=D(1^{2})=2D(1)}$, so that ${\displaystyle D(1)=0}$. Thus by ${\displaystyle K}$-linearity, ${\displaystyle D(k)=0}$ for all ${\displaystyle k\in K}$.
• If ${\displaystyle A}$ is commutative, then ${\displaystyle D(x^{2})=xD(x)+D(x)x=2xD(x)}$, and ${\displaystyle D(x^{n})=nx^{n-1}D(x)}$, by the Leibniz rule.
• More generally, for any ${\displaystyle x_{1},x_{2},\ldots ,x_{n}\in A}$, it follows by induction that

  ${\displaystyle D(x_{1}x_{2}\cdots x_{n})=\sum _{i}x_{1}\cdots x_{i-1}D(x_{i})x_{i+1}\cdots x_{n}}$

which is ${\displaystyle \textstyle \sum _{i}D(x_{i})\prod _{j\neq i}x_{j}}$ if for all ${\displaystyle i}$, ${\displaystyle D(x_{i})}$ commutes with ${\displaystyle x_{1},x_{2},\ldots ,x_{i-1}}$.

• For ${\displaystyle n>1}$, ${\displaystyle D^{n}}$ is not a derivation, instead satisfying a higher-order Leibniz rule:

${\displaystyle D^{n}(uv)=\sum _{k=0}^{n}{\binom {n}{k}}\cdot D^{nk}(u)\cdot D^{k}(v).}$

Moreover, if ${\displaystyle M}$ is an ${\displaystyle A}$-bimodule, write

${\displaystyle \operatorname {Der} _{K}(A,M)}$

for the set of ${\displaystyle K}$-derivations from ${\displaystyle A}$ to ${\displaystyle M}$.

• ${\displaystyle \mathrm {Der} _{K}(A,M)}$ is a module over ${\displaystyle K}$.
• ${\displaystyle \mathrm {Der} _{K}(A)}$ is a Lie algebra with Lie bracket defined by the commutator:

${\displaystyle [D_{1},D_{2}]=D_{1}\circ D_{2}-D_{2}\circ D_{1}.}$

since it is readily verified that the commutator of two derivations is again a derivation.

• There is an ${\displaystyle A}$-module ${\displaystyle \Omega _{A/K}}$ (called the Kähler differentials) with a ${\displaystyle K}$-derivation ${\displaystyle d:A\to \Omega _{A/K}}$ through which any derivation ${\displaystyle D:A\to M}$العوامل. أي، لأي اشتقاق${\displaystyle D}$هناك${\displaystyle A}$خريطة الوحدة النمطية${\displaystyle \varphi }$مع

${\displaystyle D:A{\stackrel {d}{\longrightarrow }}\Omega _{A/K}{\stackrel {\varphi }{\longrightarrow }}M}$

المراسلات${\displaystyle D\leftrightarrow \varphi }$هو تماثل لـ${\displaystyle A}$-الوحدات النمطية:

${\displaystyle \operatorname {Der} _{K}(A,M)\simeq \operatorname {Hom} _{A}(\Omega _{A/K},M)}$

• لو${\displaystyle k\subset K}$إذا كانت حلقة فرعية ، فـ${\displaystyle A}$يرث${\displaystyle k}$بنية الجبر، لذلك يوجد تضمين

${\displaystyle \operatorname {Der} _{K}(A,M)\subset \operatorname {Der} _{k}(A,M),}$

منذ أي${\displaystyle K}$-الاشتقاق هو من باب أولى${\displaystyle k}$-اشتقاق.

بالنظر إلى الجبر المتدرج${\displaystyle A}$وخريطة خطية متجانسة${\displaystyle D}$من الدرجة${\displaystyle |D|}$على${\displaystyle A}$،${\displaystyle D}$يكون الاشتقاق متجانساً إذا

${\displaystyle {D(ab)=D(a)b+\varepsilon ^{|a||D|}aD(b)}}$

لكل عنصر متجانس${\displaystyle A}$وكل عنصر${\displaystyle b}$ل${\displaystyle A}$لعامل التبديل${\displaystyle \varepsilon =\pm 1}$الاشتقاق المتدرج هو مجموع اشتقاقات متجانسة لها نفس${\displaystyle \varepsilon }$.

لو${\displaystyle \varepsilon =1}$، يؤول هذا التعريف إلى الحالة المعتادة. إذا${\displaystyle \varepsilon =-1}$لكن، إذن

${\displaystyle {D(ab)=D(a)b+(-1)^{|a||D|}aD(b)}}$

للفردي${\displaystyle |D|}$، و${\displaystyle D}$يُطلق عليه اسم الاشتقاق المضاد .

تشمل أمثلة المشتقات المضادة المشتق الخارجي والناتج الداخلي الذي يعمل على الأشكال التفاضلية .

الاشتقاقات المتدرجة للجبر الفائق (أي،${\displaystyle \mathbb {Z} _{2}}$تُسمى الجبر المتدرج غالبًا بالمشتقات الفائقة .

مشتقات هاس-شميت هي${\displaystyle K}$تماثلات الجبر

${\displaystyle A\to A[[t]].}$

إضافة إلى ذلك، يتم استخدام الخريطة التي ترسل سلسلة قوى رسمية.${\displaystyle \sum a_{n}t^{n}}$إلى المعامل${\displaystyle a_{1}}$يقدم اشتقاقًا.

• في الهندسة التفاضلية، تكون الاشتقاقات متجهات مماسية
• تفاضلي كاهلر
• مشتق هاس
• الاشتقاق p
• مشتقات ويرتينجر
• مشتقة الدالة الأسية

• بورباكي، نيكولاس (1989)، الجبر 1 ، عناصر الرياضيات، سبرينغر-فيرلاغ، ISBN 3-540-64243-9.
• أيزنبد، ديفيد (1999)، الجبر التبادلي مع نظرة نحو الهندسة الجبرية (الطبعة الثالثة  )، سبرينغر-فيرلاغ، ISBN 978-0-387-94269-8.
• ماتسومورا، هيديوكي (1970)، الجبر التبادلي ، سلسلة محاضرات محاضرات الرياضيات، WA Benjamin، ISBN 978-0-8053-7025-6.
• كولار، إيفان؛ سلوفاك، يان؛ ميخور، بيتر دبليو. (1993)، العمليات الطبيعية في الهندسة التفاضلية ، سبرينغر-فيرلاغ.