Mathematical notation
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus , partial differential equations and the theory of distributions , by generalising the concept of an integer index to an ordered tuple of indices.
Definition and basic properties [ edit ]
An n -dimensional multi-index is an
n
{\textstyle n}
-tuple
α
=
(
α
1
,
α
2
,
…
,
α
n
)
{\displaystyle \alpha =(\alpha _{1},\alpha _{2},\ldots ,\alpha _{n})}
of non-negative integers (i.e. an element of the
n
{\textstyle n}
-dimensional set of natural numbers , denoted
N
0
n
{\displaystyle \mathbb {N} _{0}^{n}}
).
For multi-indices
α
,
β
∈
N
0
n
{\displaystyle \alpha ,\beta \in \mathbb {N} _{0}^{n}}
and
x
=
(
x
1
,
x
2
,
…
,
x
n
)
∈
R
n
{\displaystyle x=(x_{1},x_{2},\ldots ,x_{n})\in \mathbb {R} ^{n}}
, one defines:
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 !}}}
where
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-order partial 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}},}
where
∂
i
α
i
:=
∂
α
i
/
∂
x
i
α
i
{\displaystyle \partial _{i}^{\alpha _{i}}:=\partial ^{\alpha _{i}}/\partial x_{i}^{\alpha _{i}}}
(see also 4-gradient ). Sometimes the notation
D
α
=
∂
α
{\displaystyle D^{\alpha }=\partial ^{\alpha }}
is also used.[ 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}}
(or
R
n
{\displaystyle \mathbb {R} ^{n}}
),
α
,
ν
∈
N
0
n
{\displaystyle \alpha ,\nu \in \mathbb {N} _{0}^{n}}
, and
f
,
g
,
a
α
:
C
n
→
C
{\displaystyle f,g,a_{\alpha }\colon \mathbb {C} ^{n}\to \mathbb {C} }
(or
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 }.}
Note that, since x y is a vector and α is a multi-index, the expression on the left is short for (x 1 y 1 )α 1 ⋯(x n y n )α n .
Leibniz formula
For smooth functions
f
{\textstyle f}
and
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 an analytic function
f
{\textstyle f}
in
n
{\textstyle n}
variables one has
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 }}.}
In fact, for a smooth enough function, we have the similar 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),}
where the last term (the remainder) depends on the exact version of Taylor's formula. For instance, for the Cauchy formula (with integral remainder), one gets
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 linear partial differential operator
A formal linear
N
{\textstyle N}
-th order partial differential operator in
n
{\textstyle n}
variables is written as
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 with compact support in a bounded domain
Ω
⊂
R
n
{\displaystyle \Omega \subset \mathbb {R} ^{n}}
one has
∫
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}.}
This formula is used for the definition of distributions and weak derivatives .
If
α
,
β
∈
N
0
n
{\displaystyle \alpha ,\beta \in \mathbb {N} _{0}^{n}}
are multi-indices and
x
=
(
x
1
,
…
,
x
n
)
{\displaystyle x=(x_{1},\ldots ,x_{n})}
, then
∂
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 the power rule for the ordinary derivative ; if α and β are in
{
0
,
1
,
2
,
…
}
{\textstyle \{0,1,2,\ldots \}}
, then
d
α
d
x
α
x
β
=
{
β
!
(
β
−
α
)
!
x
β
−
α
if
α
≤
β
,
0
otherwise.
{\displaystyle {\frac {d^{\alpha }}{dx^{\alpha }}}x^{\beta }={\begin{cases}{\frac {\beta !}{(\beta -\alpha )!}}x^{\beta -\alpha }&{\hbox{if}}\,\,\alpha \leq \beta ,\\0&{\hbox{otherwise.}}\end{cases}}}
(1 )
Suppose
α
=
(
α
1
,
…
,
α
n
)
{\displaystyle \alpha =(\alpha _{1},\ldots ,\alpha _{n})}
,
β
=
(
β
1
,
…
,
β
n
)
{\displaystyle \beta =(\beta _{1},\ldots ,\beta _{n})}
, and
x
=
(
x
1
,
…
,
x
n
)
{\displaystyle x=(x_{1},\ldots ,x_{n})}
. Then we have that
∂
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 each
i
{\textstyle i}
in
{
1
,
…
,
n
}
{\textstyle \{1,\ldots ,n\}}
, the function
x
i
β
i
{\displaystyle x_{i}^{\beta _{i}}}
only depends on
x
i
{\displaystyle x_{i}}
. In the above, each partial differentiation
∂
/
∂
x
i
{\displaystyle \partial /\partial x_{i}}
therefore reduces to the corresponding ordinary differentiation
d
/
d
x
i
{\displaystyle d/dx_{i}}
. Hence, from equation (1 ), it follows that
∂
α
x
β
{\displaystyle \partial ^{\alpha }x^{\beta }}
vanishes if
α
i
>
β
i
{\textstyle \alpha _{i}>\beta _{i}}
for at least one
i
{\textstyle i}
in
{
1
,
…
,
n
}
{\textstyle \{1,\ldots ,n\}}
. If this is not the case, i.e., if
α
≤
β
{\textstyle \alpha \leq \beta }
as multi-indices, then
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}}}
for each
i
{\displaystyle i}
and the theorem follows. 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 on PlanetMath , which is licensed under the Creative Commons Attribution/Share-Alike License .