Mathematical notation
Multi-index notation is amathematical notation that simplifies formulas used inmultivariable calculus ,partial differential equations and the theory ofdistributions , by generalising the concept of an integerindex to an orderedtuple of indices.
Definition and basic properties [ edit ] Ann -dimensionalmulti-index is ann {\textstyle n} tuple
α = ( α 1 , α 2 , … , α n ) {\displaystyle \alpha =(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n})} ofnon-negative integers (i.e. an element of then {\textstyle n} dimensional set ofnatural numbers , denotedN 0 n {\displaystyle \mathbb {N} _{0}^{n}}
For multi-indicesα , β ∈ N 0 n {\displaystyle \alpha ,\beta \in \mathbb {N} _{0}^{n}} x = ( x 1 , x 2 , … , x n ) ∈ R n {\displaystyle x=(x_{1},x_{2},\ldots ,x_{n})\in \mathbb {R} ^{n}}
Componentwise sum and difference α ± β = ( α 1 ± β 1 , α 2 ± β 2 , … , α n ± β n ) {\displaystyle \alpha \pm \beta =(\alpha _{1}\pm \beta _{1},\,\alpha _{2}\pm \beta _{2},\ldots ,\,\alpha _{n}\pm \beta _{n})} Partial order α ≤ β ⇔ α i ≤ β i ∀ i ∈ { 1 , … , n } {\displaystyle \alpha \leq \beta \quad \Leftrightarrow \quad \alpha _{i}\leq \beta _{i}\quad \forall \,i\in \{1,\ldots ,n\}} Sum of components (absolute value) | α | = α 1 + α 2 + ⋯ + α n {\displaystyle |\alpha |=\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}} Factorial α ! = α 1 ! ⋅ α 2 ! ⋯ α n ! {\displaystyle \alpha !=\alpha _{1}!\cdot \alpha _{2}!\cdots \alpha _{n}!} Binomial coefficient ( α β ) = ( α 1 β 1 ) ( α 2 β 2 ) ⋯ ( α n β n ) = α ! β ! ( α − β ) ! {\displaystyle {\binom {\alpha }{\beta }}={\binom {\alpha _{1}}{\beta _{1}}}{\binom {\alpha _{2}}{\beta _{2}}}\cdots {\binom {\alpha _{n}}{\beta _{n}}}={\frac {\alpha !}{\beta !(\alpha -\beta )!}}} Multinomial coefficient ( k α ) = k ! α 1 ! α 2 ! ⋯ α n ! = k ! α ! {\displaystyle {\binom {k}{\alpha }}={\frac {k!}{\alpha _{1}!\alpha _{2}!\cdots \alpha _{n}!}}={\frac {k!}{\alpha !}}} k := | α | ∈ N 0 {\displaystyle k:=|\alpha |\in \mathbb {N} _{0}} Power x α = x 1 α 1 x 2 α 2 … x n α n {\displaystyle x^{\alpha }=x_{1}^{\alpha _{1}}x_{2}^{\alpha _{2}}\ldots x_{n}^{\alpha _{n}}} Higher-orderpartial derivative ∂ α = ∂ 1 α 1 ∂ 2 α 2 … ∂ n α n , {\displaystyle \partial ^{\alpha }=\partial _{1}^{\alpha _{1}}\partial _{2}^{\alpha _{2}}\ldots \partial _{n}^{\alpha _{n}},} ∂ i α i := ∂ α i / ∂ x i α i {\displaystyle \partial _{i}^{\alpha _{i}}:=\partial ^{\alpha _{i}}/\partial x_{i}^{\alpha _{i}}} 4-gradient ). Sometimes the notationD α = ∂ α {\displaystyle D^{\alpha }=\partial ^{\alpha }} [ 1] The multi-index notation allows the extension of many formulae from elementary calculus to the corresponding multi-variable case. Below are some examples. In all the following,x , y , h ∈ C n {\displaystyle x,y,h\in \mathbb {C} ^{n}} R n {\displaystyle \mathbb {R} ^{n}} α , ν ∈ N 0 n {\displaystyle \alpha ,\nu \in \mathbb {N} _{0}^{n}} f , g , a α : C n → C {\displaystyle f,g,a_{\alpha }\colon \mathbb {C} ^{n}\to \mathbb {C} } R n → R {\displaystyle \mathbb {R} ^{n}\to \mathbb {R} }
Multinomial theorem ( ∑ i = 1 n x i ) k = ∑ | α | = k ( k α ) x α {\displaystyle \left(\sum _{i=1}^{n}x_{i}\right)^{k}=\sum _{|\alpha |=k}{\binom {k}{\alpha }}\,x^{\alpha }} Multi-binomial theorem ( x + y ) α = ∑ ν ≤ α ( α ν ) x ν y α − ν . {\displaystyle (x+y)^{\alpha }=\sum _{\nu \leq \alpha }{\binom {\alpha }{\nu }}\,x^{\nu }y^{\alpha -\nu }.} x +y α (x 1 +y 1 )α 1 x n y n α n .Leibniz formula For smooth functionsf {\textstyle f} g {\textstyle g} ∂ α ( f g ) = ∑ ν ≤ α ( α ν ) ∂ ν f ∂ α − ν g . {\displaystyle \partial ^{\alpha }(fg)=\sum _{\nu \leq \alpha }{\binom {\alpha }{\nu }}\,\partial ^{\nu }f\,\partial ^{\alpha -\nu }g.} Taylor series For ananalytic function f {\textstyle f} n {\textstyle n} f ( x + h ) = ∑ α ∈ N 0 n ∂ α f ( x ) α ! h α . {\displaystyle f(x+h)=\sum _{\alpha \in \mathbb {N} _{0}^{n}}{{\frac {\partial ^{\alpha }f(x)}{\alpha !}}h^{\alpha }}.} Taylor expansion f ( x + h ) = ∑ | α | ≤ n ∂ α f ( x ) α ! h α + R n ( x , h ) , {\displaystyle f(x+h)=\sum _{|\alpha |\leq n}{{\frac {\partial ^{\alpha }f(x)}{\alpha !}}h^{\alpha }}+R_{n}(x,h),} R n ( x , h ) = ( n + 1 ) ∑ | α | = n + 1 h α α ! ∫ 0 1 ( 1 − t ) n ∂ α f ( x + t h ) d t . {\displaystyle R_{n}(x,h)=(n+1)\sum _{|\alpha |=n+1}{\frac {h^{\alpha }}{\alpha !}}\int _{0}^{1}(1-t)^{n}\partial ^{\alpha }f(x+th)\,dt.} General linearpartial differential operator A formal linearN {\textstyle N} n {\textstyle n} P ( ∂ ) = ∑ | α | ≤ N a α ( x ) ∂ α . {\displaystyle P(\partial )=\sum _{|\alpha |\leq N}{a_{\alpha }(x)\partial ^{\alpha }}.} Integration by parts For smooth functions withcompact support in a bounded domainΩ ⊂ R n {\displaystyle \Omega \subset \mathbb {R} ^{n}} ∫ Ω u ( ∂ α v ) d x = ( − 1 ) | α | ∫ Ω ( ∂ α u ) v d x . {\displaystyle \int _{\Omega }u(\partial ^{\alpha }v)\,dx=(-1)^{|\alpha |}\int _{\Omega }{(\partial ^{\alpha }u)v\,dx}.} distributions andweak derivatives . Ifα , β ∈ N 0 n {\displaystyle \alpha ,\beta \in \mathbb {N} _{0}^{n}} x = ( x 1 , … , x n ) {\displaystyle x=(x_{1},\ldots ,x_{n})} ∂ α x β = { β ! ( β − α ) ! x β − α if α ≤ β , 0 otherwise. {\displaystyle \partial ^{\alpha }x^{\beta }={\begin{cases}{\frac {\beta !}{(\beta -\alpha )!}}x^{\beta -\alpha }&{\text{if}}~\alpha \leq \beta ,\\0&{\text{otherwise.}}\end{cases}}}
The proof follows from thepower rule for theordinary derivative ; ifα andβ are in{ 0 , 1 , 2 , … } {\textstyle \{0,1,2,\ldots \}}
Supposeα = ( α 1 , … , α n ) {\displaystyle \alpha =(\alpha _{1},\ldots ,\alpha _{n})} β = ( β 1 , … , β n ) {\displaystyle \beta =(\beta _{1},\ldots ,\beta _{n})} x = ( x 1 , … , x n ) {\displaystyle x=(x_{1},\ldots ,x_{n})} ∂ α x β = ∂ | α | ∂ x 1 α 1 ⋯ ∂ x n α n x 1 β 1 ⋯ x n β n = ∂ α 1 ∂ x 1 α 1 x 1 β 1 ⋯ ∂ α n ∂ x n α n x n β n . {\displaystyle {\begin{aligned}\partial ^{\alpha }x^{\beta }&={\frac {\partial ^{\vert \alpha \vert }}{\partial x_{1}^{\alpha _{1}}\cdots \partial x_{n}^{\alpha _{n}}}}x_{1}^{\beta _{1}}\cdots x_{n}^{\beta _{n}}\\&={\frac {\partial ^{\alpha _{1}}}{\partial x_{1}^{\alpha _{1}}}}x_{1}^{\beta _{1}}\cdots {\frac {\partial ^{\alpha _{n}}}{\partial x_{n}^{\alpha _{n}}}}x_{n}^{\beta _{n}}.\end{aligned}}}
For eachi {\textstyle i} { 1 , … , n } {\textstyle \{1,\ldots ,n\}} x i β i {\displaystyle x_{i}^{\beta _{i}}} x i {\displaystyle x_{i}} ∂ / ∂ x i {\displaystyle \partial /\partial x_{i}} d / d x i {\displaystyle d/dx_{i}} 1 ∂ α x β {\displaystyle \partial ^{\alpha }x^{\beta }} α i > β i {\textstyle \alpha _{i}>\beta _{i}} i {\textstyle i} { 1 , … , n } {\textstyle \{1,\ldots ,n\}} α ≤ β {\textstyle \alpha \leq \beta } d α i d x i α i x i β i = β i ! ( β i − α i ) ! x i β i − α i {\displaystyle {\frac {d^{\alpha _{i}}}{dx_{i}^{\alpha _{i}}}}x_{i}^{\beta _{i}}={\frac {\beta _{i}!}{(\beta _{i}-\alpha _{i})!}}x_{i}^{\beta _{i}-\alpha _{i}}} i {\displaystyle i} Q.E.D.
^ Reed, M.; Simon, B. (1980).Methods of Modern Mathematical Physics: Functional Analysis I (Revised and enlarged ed.). San Diego: Academic Press. p. 319.ISBN 0-12-585050-6 Saint Raymond, Xavier (1991).Elementary Introduction to the Theory of Pseudodifferential Operators . Chap 1.1 . CRC Press.ISBN 0-8493-7158-9 This article incorporates material from multi-index derivative of a power onPlanetMath , which is licensed under theCreative Commons Attribution/Share-Alike License .
