Title: Theory / Important / Math / Analysis / Zorich{ / 06 Integration / Multiple variable
Tags: closed récupéré Wrong_taylor_series fromMath hasSeenConjDef mathToSee mid1516974248815 toUnary
On every note of this deck, unless a note says otherwise:
Chapter: 11

Definition: \(\inf_{P}S(f,p)\)
Page: 117
Chapter: 11
Section: 1.3.b
Variables: \(f:[a,b]\to\mathbb R\)
Notation: \(\overline {\mathcal J}\)
Index: 11, 2
Kind: Def, Theo
Definition2: \(\lim_{\lambda(P)\to 0}S(f,P)\) assuming \(f\) is bounded

Definition: \(\lim_{\lambda(P)\to 0}S(f;P)\)
Page: 341
Chapter: 6
Section: 1
Variables: \(n\in\mathbb N\),
\(I\subseteq\mathbb R^n\) an interval
\(f:I\to\mathbb R\)
Notation: \(\overline I\)
Index: 3
Kind: Prop

Definition: \(\sup_{P}s(f;P)\)
Page: 117
Chapter: 11
Section: 1.3.b
Variables: \(f:[a,b]\to\mathbb R\)
Notation: \(\underline{\mathcal J}\)
Index: 11, 2
Kind: Def, theo
Definition2: \(\lim_{\lambda(P)\to 0}s(f,P)\) assuming \(f\) is bounded

Name: An interval of \(\mathbb R^n\)
Definition: A cartesian product of \(n\) intervals
Page: 109
Chapter: 11
Section: 1
Variables: \(n\in\mathbb N\)
Kind: Def
Name2: A coordinate parallelepiped in \(\mathbb R^n\)
Denoted by: \(I\)

Definition: \(\Pi [a_i,b_i]\)
Page: 109
Chapter: 11
Section: 1
Variables: \(n\in\mathbb N\),
\(a=(a^1,\dots,a^n)\) and \(b=(b^1,\dots,b^n)\)
Notation: \(I_{a,b}\)
Kind: Def
Notation2: \(a\le x\le b\)

Name: The volume of \(I_{a,b}\)
Definition: \(\Pi_i (b_i-a_i)\)
Page: 109
Chapter: 11
Section: 1
Variables: \(n\in\mathbb N\),
\(a=(a^1,\dots,a^n)\)
\(b=(b^1,\dots,b^n)\)
Notation: [$]\abs{I_{a,b}}[/$]
Index: 2
Kind: Def
Name2: The measure of \(I_{a,b}\)

Question: [$]\abs{I_{\lambda a,\lambda b}}[/$]
Answer: equals [$]\lambda^n\abs{I_{a,b}}[/$]
Page: 109
Chapter: 11
Section: 1
Variables: \(n\in\mathbb N\),
\(a=(a^1,\dots,a^n)\) and \(b=(b^1,\dots,b^n)\),
\(\lambda\in\mathbb R\)
Context: Multidimensional intervals
Index: 1a
Kind: Lem

Smaller0: \(I\) is the union of the \(I_i\)'s
Greater0: [$]\abs{I}=\sum\abs{I_i}[/$]
Page: 109
Section: 1
Variables: \(n\in\mathbb N\),
\(I,I_1,\dots, I_n\) parallelepipeds
Index: 1b
Kind: Lem
Smaller1: the intersections of the \(I_i\)'s are included in their bordure

Smaller0: \(I\subseteq\bigcup_{i}I_i\)
Greater0: [$]\abs{I}\le\sum\abs{I_i}[/$]
Page: 109
Section: 1
Variables: \(n\in\mathbb N\),
\(I,I_1,\dots, I_n\) parallelepipeds
Index: 1c
Kind: Lem
Counterexample: \(I=I_1=I_2\)

Name: A partition of \(I\)
Definition: The cartesian products of the elements of partitions of each \([a^i,b^i]\)
Page: 110
Chapter: 11
Section: 1.1.b
Variables: \(n\in\mathbb N\),
\(I_{a,b}\) a parallelepiped
Index: 3
Kind: Def
Denoted by: \(P\)

Name: The mesh of \(P\)
Definition: The volume of the greatest element of \(P\)
Page: 110
Chapter: 11
Section: 1.1.b
Variables: \(n\in\mathbb N\),
\(I_{a,b}\) a parallelepiped
\(P\) a partition of \(I_{a,b}\)
Notation: \(\lambda(P)\)
Index: 3
Kind: Def

Name: A partition with distingished point
Definition: A pair with a partition, and a point in each element of the partition
Page: 110
Chapter: 11
Section: 1.1.b
Variables: \(n\in\mathbb N\),
\(I_{a,b}\) a parallelepiped
Index: 5
Kind: Def
Denoted by: \((P,\xi)\)

Definition: The set of partitions of \(I_{a,b}\) with distinguished point whose mesh is at most \(d\)
Page: 110
Chapter: 11
Section: 1.1.b
Variables: \(n\in\mathbb N\),
\(I_{a,b}\) a parallelepiped,
\(d>0\)
Notation: \(B_d\)
Index: 5
Kind: Def

Definition: The base containing, for each \(\epsilon>0\), the set \(B_d\) of distinguished partitions of mesh at most \(d\)
Page: 110
Chapter: 11
Section: 1.1.b
Variables: \(n\in\mathbb N\),
\(I_{a,b}\) a parallelepiped,
\(d>0\)
Notation: The base \(\lambda(P)\to 0\)
Index: 5
Kind: Def
Extra: \(P\) is a variable representing partitions

Smaller0: [$]\int_If(x^1,\dots, x^n)\dd x^1,\dots,dx^n[/$]
Greater0: [$]\int_{a_1}^{b_1}\dots \int_{a_n}^{b_n}f(x^1,\dots, x^n)\dd x^1,\dots,dx^n[/$]
Page: 111
Section: 1.1 d
Variables: \(\mathbb F\) a real vector space
\(I=\prod_{i=1}^n[a_i,b_i]\) a parallelepiped,
\(f:I\to\mathbb F\)
Smaller to greater: =

Name: A degenerate interval
Definition: An interval \(I_{a,b}\) with some \(a^i\) equals to \(b^i\)
Page: 112
Chapter: 11
Section: 1.1 d
Context: \(\mathbb R^n\)
Index: 2
Kind: Lem

Smaller0: \(E\) is compact
Greater0: \(E\) is of measure 0
Page: 113
Section: 1
Variables: \(n\in\mathbb N\),
\(E\subseteq\mathbb R^n\)
Index: 3b
Kind: Lem
Greater1: \(E\) is covered by a FINITE number of intervals whose sum of measure is less than \(\epsilon\) for \(\epsilon>0\) arbitrary
Connect Greater0: iff

Smaller0: \(E\) is compact
Greater0: for every \(\epsilon{\gt}0\) there exists \(\delta{\gt}0\)
Page: 113
Section: 1 b
Variables: \(n\in\mathbb N\),
\(E\subseteq\mathbb R^n\),
\(f:E\to\mathbb R\),
\(\omega_0{\gt}0\)
\(x\in E\)
Index: 4
Kind: Lem
Smaller1: \(\omega(f,x)\le\omega_0\)
Greater1: \(\omega(f;U_K^\delta(x))<\omega_0+\epsilon\) for each \(x\in E\)
Connect Greater0: such that
Extra: locally compact ?

Smaller0: \(f\) is Riemann integrable on \(E\)
Greater0: \(f\) is bounded
Page: 113
Section: 1 c
Variables: \(n\in\mathbb N\),
\(E\subseteq\mathbb R^n\)
\(f:E\to\mathbb R\),
Index: 1
Kind: Theo
Smaller to greater: iff
Assuming: \(E\) is admissible
Name: Lebesgue's criterion
Greater1: \(f\) is continuous almost everywhere on \(E\)
Extra: relire preuve

Name: An admissible set
Definition: bounded,
Page: 119, 122
Chapter: 11
Section: 2.1
Context: \(\mathbb R^n\)
Index: 1, 1, 2
Kind: Def, Ex
Definition2: with boundary of measure 0 in the sense of Lebesgue or of Jordan
Example: Cube, Tetrahedron, ball
Conjdef: A subset of \(\mathbb R^n\),
Name2: A Jordan-measurable set
Example2: Given \(I\subseteq\mathbb R^n\) an interval, \(\phi_1<\phi_2:I\to\mathbb R\), \(\{(x,y)\in I\times\mathbb R\mid \phi_1(x)\le y\le \phi_2(x)\}\)

Name: The boundary of \(E\)
Definition: The set of \(x\in\mathbb R^n\) whose every neighbor intersects \(E\) and \(\mathbb R^n\setminus E\)
Page: 119
Chapter: 11
Section: 2.1
Variables: \(E\subseteq\mathbb R^n\)
Notation: \(\partial E\)
Kind: Def, Ex
Typ: closed

Smaller0: \(\partial(E_1\cup E_2)\)
Greater0: \(\partial(E_1)\cup\partial(E_2)\)
Page: 119
Section: 2.1
Variables: \(E_1,E_2\subseteq\mathbb R^n\)
Kind: Def, Ex
Smaller to greater: \(\subseteq\)
Greater to smaller: \(\supseteq\)
Hide sides: yes

Smaller0: \(\partial(E_1\cap E_2)\)
Greater0: \(\partial(E_1)\cup\partial(E_2)\)
Page: 119
Section: 2.1
Variables: \(E_1,E_2\subseteq\mathbb R^n\)
Kind: Def, Ex
Smaller to greater: \(\subseteq\)
Greater to smaller: \(\supseteq\)
Hide sides: yes

Smaller0: \(\partial(E_1\setminus E_2)\)
Greater0: \(\partial(E_1)\cup\partial(E_2)\)
Page: 119
Section: 2.1
Variables: \(E_1,E_2\subseteq\mathbb R^n\)
Kind: Def, Ex
Smaller to greater: \(\subseteq\)
Greater to smaller: \(\supseteq\)
Hide sides: yes

Name: Admissible sets
Page: 120
Section: 2.1
Chapter: 11
Index: 2
Variables: Set bounded with boundary of neasure 0
Kind: Lem
Under: Boolean operation
Condition: finite, but not infinite
Not2: not
Under2: complement

Smaller0: continuous almost everywhere
Greater0: \(E\) is admissible
Page: 120
Section: 2.1
Variables: \(E\subseteq\mathbb R^n\)
Index: 2
Kind: Lem
Smaller to greater: iff
Prefix Smaller: \(\Xi_E\) is

Definition: [$]\defFun{\mathbb R^n}{V}{x}{f(x)\text{ if $x\in E$, } 0\text{ otherwise}}[/$]
Page: 120
Chapter: 11
Section: 2.2
Variables: \(E\subseteq\mathbb R^n\),
\(V\) a set containing a zero
\(f:E\to V\)
Notation: \(f_{\Xi_E}\)
Index: 2
Kind: Def

Definition: [$]\int_{I\subseteq E}f_{\Xi_E}(x)\dd x[/$] for \(I\) any interval containing \(E\)
Page: 120, 121
Chapter: 11
Section: 2.2
Variables: \(E\subseteq\mathbb R^n\),
\(f:E\to\mathbb R^n\)
Notation: [$]\int_Ef(x)\dd x[/$]
Index: 2, 3
Kind: Def, Lem
Typ: Independant of \(I\)

Name: A Riemann integrable function over \(E\)
Definition: A function such that [$]\int_Ef(x)\dd x[/$] is defined, for the Riemann definition of integral
Page: 120
Chapter: 11
Section: 2.2
Variables: \(E\subseteq\mathbb R^n\),
Index: 2
Kind: Def

Name: A Riemann nonintegrable function over \(E\)
Definition: A function such that [$]\int_Ef(x)\dd x[/$] is not defined
Page: 120
Chapter: 11
Section: 2.2
Variables: \(E\subseteq\mathbb R^n\),
Index: 2
Kind: Def
Example: Dirichlet functions (characteristic function of \(\mathbb Q\))

Name: The Jordan measure of content of \(E\)
Definition: [$]\int_E1\dd x[/$]
Page: 121, https://en.wikipedia.org/wiki/Jordan_measure
Chapter: 11
Section: 23
Variables: \(E\subseteq\mathbb R^n\),
Notation: \(\mu(E)\)
Index: 3
Kind: Def
Name2: The length of \(E\) (assuming \(n=1\))
Assuming: \(E\) admissible
Name3: The area of \(E\) (assuming \(n=2\))

Name: A set of measure zero in the sense of Jordan
Definition: A set \(E\subseteq\mathbb R^n\) such that, for every \(\epsilon>0\), there exists a finite cover of \(E\) by intervals whose sum of measure is at most \(\epsilon\)
Page: 122
Chapter: 11
Section: 23
Index: 3
Kind: Def
Name2: A set of content zero

Smaller0: \(X\) is of measure zero in the sense of Jordan
Greater0: \(X\) is of measure zero in the sense of Lebesgue
Page: 122
Section: 2.3
Variables: \(n\in\mathbb N\)
\(X\subseteq \mathbb R^n\)
Counterexample: \(\mathbb Q\) can't be \(\epsilon\)-covered by finitely many intervals

Definition: [$]\defFun{\mathcal R(E)}{\mathbb R}{f}{\int_E f}[/$]
Page: 124
Section: 3.1
Chapter: 11
Variables: \(n\in\mathbb N\)
\(E\subseteq\mathbb R^n\)
Linear: Linear functional

Definition: The quotient of \({\mathcal R(E)}\), under the equivalence relation of function equals on almost all points
Page: 124
Chapter: 11
Section: 3.1
Variables: \(n\in\mathbb N\)
\(E\subseteq\mathbb R^n\)
Notation: \(\widetilde{\mathcal R(E)}\)
Property: Normed vector space

Definition: \(\int g\), for \(g\) in the equivalence class \(f\)
Page: 124
Chapter: 11
Section: 3.1
Variables: \(n\in\mathbb N\)
\(E\subseteq\mathbb R^n\)
\(f\in\widetilde{\mathcal R(E)}\)
Context: \(\widetilde{\mathcal R(E)}\)
Notation: \(\int f\)

Definition: [$]\int_E\abs{f}[/$]
Page: 127
Chapter: 11
Section: 3.1
Variables: \(n\in\mathbb N\)
\(E\subseteq\mathbb R^n\)
\(f\in\widetilde{\mathcal R(E)}\)
Notation: [$]\norm{f}[/$]

Smaller0: \(f\) is integrable on \(E_1\cup E_2\)
Greater0: \(f\) is integrable on \(E_1\) and on \(E_2\)
Page: 124
Section: 3.1
Variables: \(n\in\mathbb N\)
\(E_1,E_2\subseteq\mathbb R^n\)
\(f:\mathbb R^n\to \mathbb R\)
Smaller to greater: iff
Assuming: \(E_1, E_2\) admissible

Smaller0: \(f\) is integrable on \(E_1\cup E_2\) (i.e. \(f\) is integrable on \(E_1\) and on \(E_2\))
Greater0: \(f\) is integrable on \(E_1\cap E_2\)
Page: 124
Section: 3.2
Variables: \(n\in\mathbb N\)
\(E_1,E_2\subseteq\mathbb R^n\)
\(f:\mathbb R^n\to\mathbb R\)
Assuming: \(E_1, E_2\) admissible
Construction: \(E_1,E_2\) constructible, thus their intersection constructible
It then suffices to use the definition «integrable»=«bounded and continuous on almost all points»
Counterexample: \(f:\mathcal X_{\mathbb Q\setminus[0,1]}\), \(E_1=\{x\ge0\}\) and \(E_2=\{x\le 1\}\)

Smaller0: \(f\) is integrable on \(E_1\cup E_2\) (i.e. \(f\) is integral on \(E_1\) and on \(E_2\))
Greater0: [$]\int_{E_1\cup E_2}f(x)\dd x[/$]
Page: 124
Section: 3.2
Variables: \(n\in\mathbb N\)
\(E_1,E_2\subseteq\mathbb R^n\)
\(f:\mathbb R^n\to \mathbb R\)
Assuming: \(E_1, E_2\) admissible
Smaller1: \(\mu(E_1\cap E_2)=0\)
Greater1: [$]\int_{E_1}f(x)\dd x+\int_{E_2}f(x)\dd x[/$]
Connect Greater0: =

Smaller0: \(f\) integrable
Greater0: \(\ge0\)
Page: 125
Section: 3.3b
Variables: \(n\in\mathbb N\)
\(E\subseteq \mathbb R^n\)
\(f\in\mathcal R(E)\)
Index: 4
Kind: Prop
Prefix Greater: \(\int_E{f}\)
Assuming: \(E\) admissible,
Smaller1: \(f\) nonnegative on \(E\)

Smaller0: \(f,g\) integrable
Greater0: \(\int_E f\le \int_E g\)
Page: 125
Section: 3.3b
Variables: \(n\in\mathbb N\)
\(E\subseteq \mathbb R^n\)
\(f,g\in\mathcal R(E)\)
Index: 1
Kind: Cor
Assuming: \(E\) admissible,
Smaller1: \(f\le g\) on \(E\)

Smaller0: \(f\) integrable
Greater0: \(m\mu(E)\le\int_E f\le M\mu (E)\)
Page: 125
Section: 3.3b
Variables: \(n\in\mathbb N\)
\(E\subseteq \mathbb R^n\)
\(f\in\mathcal R(E)\)
\(M,m\in\mathbb M\)
Index: 2
Kind: Cor
Assuming: \(E\) admissible,
Smaller1: \(m\le f\le M\) on \(E\)

Question: \(\frac{\int f}{\mu(E)}\) belongs to
Answer: \([\inf_{x\in E}f(x),\sup_{x\in E}f(x)]\)
Page: 125
Chapter: 11
Section: 3.3b
Variables: \(n\in\mathbb N\)
\(E\subseteq \mathbb R^n\)
\(f\in\mathcal R(E)\)
Index: 3
Kind: Cor
Assuming: \(E\) admissible,

Smaller0: \(E\) connected
Greater0: is of the form \(f(\xi)\) for some \(\xi\in E\)
Page: 125
Section: 3.3b
Variables: \(n\in\mathbb N\)
\(E\subseteq \mathbb R^n\)
\(f\in\mathcal R(E)\)
Index: 4
Kind: Cor
Prefix Greater: \(\frac{\int f}{\mu(E)}\)
Assuming: \(E\) admissible,

Smaller0: \(m\le f\le M\)
Greater0: \(m\int_Eg\le \int_E fg\le M\int_E g\)
Page: 125
Section: 3.3b
Variables: \(n\in\mathbb N\)
\(E\subseteq \mathbb R^n\)
\(f,g\in\mathcal R(E)\)
\(m,M\in\mathbb R\)
Index: 5
Kind: Cor
Assuming: \(E\) admissible,
Smaller1: \(g\ge 0\)

Smaller0: \(f\ge0\)
Greater0: \(f=0\) almost everywhere
Page: 127
Section: 3.3b
Variables: \(n\in\mathbb N\)
\(E\subseteq \mathbb R^n\)
\(f\in\mathcal R(E)\)
Kind: Lem
Smaller1: \(\int f=0\)
Smaller2: \(E\) admissible,

Name: A multiple integral
Definition: An integration done over multiple variable simultaneously
Page: 111
Chapter: 11

Name: Iterated integration of \(f\)
Definition: Finding a closed/manageable form/bounds for [$]F(x)=\int_X f(x,y)\dd x[/$], and then computing [$]\int_YF(y)\dd y[/$]
Page: 129
Chapter: 11
Variables: \(X,Y\) measurable spaces
\(f:X\times Y\to\mathbb R\)

Smaller0: \(f\in\mathcal R(X\times Y)\)
Greater0: defined almost everywhere
Page: 131
Section: 4.2
Variables: \(X,Y\) measurable spaces
\(f:X\times Y\to\mathbb R\)
Index: 1
Kind: Cor
Prefix Greater: [$]x\mapsto_Y f(x,y)\dd y[/$] is

Smaller0: [$]\int_{\prod_i[a_i, b_i]}f(x)\dd x[/$]
Greater0: [$]\int_{[a_1, b_1]}\dd x^1\dots\int_{[a_n,b_n]}\dd x^nf(x)[/$]
Page: 131
Section: 4.2
Variables: \((a_i<b_i)_{i=1}^n\)
\(f:\prod_i[a_i,b_i]\to\mathbb R\)
Index: 2
Kind: Cor
Smaller to greater: =

Smaller0: [$]\int_{E}f(x)\dd x[/$]
Greater0: [$]\int_{D}\dd x\int_{\phi_1(x)}^{\phi_2(x)}(x,y)\dd y[/$]
Page: 131
Section: 4.2
Variables: \(D\subseteq\mathbb R^n\) bounded
\(\phi_1,\phi_2:X\to\mathbb R\)
\(E=\{(x,y)\in\mathbb R^n\mid x\in D\land \phi_1(x)\le y\le\phi_2(x)\}\)
Index: 3
Kind: Cor
Smaller to greater: =

Name: Jordan-measurable set
Page: 131
Section: 4.2
Chapter: 11
Index: 3
Kind: Cor
Under: transforming \(D\) into \(\{(d,y)\mid d\in D, \phi_1(d)\le y\le \phi_2(d)\}\) for \(\phi_1,\phi_2\) continous and bounded
Under2: union?
Under3: intersection?
Under4: product?
Under5: suberset?

Smaller0: the measure of \(\{(x,y)\in\mathbb R^n\mid x\in D\land \phi_1(x)\le y\le\phi_2(x)\}\)
Greater0: [$]\int_D(\phi_2(x)-\phi_1(x))\dd x[/$]
Page: 133
Section: 4.2
Variables: \(D\subseteq\mathbb R^n\) bounded
\(\phi_1,\phi_2:X\to\mathbb R\)
Index: 4
Kind: Cor
Smaller to greater: equals
Assuming: \(\phi_1,\phi_2\) continuous

Smaller0: \(E\) is bounded
Greater0: measurable for almost every \(x_1\)
Page: 133
Section: 4.2
Variables: \(E\subseteq\mathbb R^n\)
Index: 4
Kind: Cor
Prefix Greater: \(E_{x_1}=\{(x_2,\dots, x_n)\mid (x_1,x_2,\dots, x_n)\in E)\) is
Smaller1: \(E\) is measurable

Question: The measure of an hypersphere of dimension \(m\) and radius \(r\) is
Answer: \(2\frac{(2\pi)^{\lfloor \frac k2\rfloor}}{(m)!!}r^{m}\) with \(0!!=1!!=1\) and \(n!!=n\times (n-2)!!\)
Page: 134
Chapter: 11
Section: 4.2
Index: 3
Kind: Ex

Definition: \(\prod_{i=1}^{\left\lfloor\frac n2\right\rfloor}(n-2i)\)
Page: 134
Chapter: 11
Section: 4.2
Variables: \(n\in\mathbb N\)
Notation: \(n!!\)
Index: 3
Kind: Ex
Intuition: Similar to factorial, but with step 2