Title: Theory / Important / Math / Analysis / Zorich{ / 06 Integration / Single variable
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Question: How to approximate \(s(t_n)\), given \(s(t_0)\), \(v(t_i)\) and \(\Delta t_i\)
Answer: \(s(t_0)+\sum v(t_i)\Delta t_i\)
Page: 331
Chapter: 6
Section: 1.1
Variables: \((t_0,\dots, t_n)\in\mathbb R\) increasing
\(s:\mathbb R\to\mathbb R\),
\(v=s'\)
\(s(t_0)\) and \(v(t_i)\) for \(i>0\) given

Definition: \(t_i-t_{i-1}\)
Page: 331
Chapter: 6
Section: 1.1
Variables: \((t_0,\dots, t_n)\in\mathbb R\) increasing
Notation: \(\Delta t_i\)

Smaller0: \(\lim_{\max{\Delta t_i}\to 0}\sum_{i=1}^ns'(\tau_i)\Delta t_i\)
Greater0: \(s(t)-s(t_0)\)
Page: 331
Chapter: 6
Section: 1.1
Variables: \(s:\mathbb R\to\mathbb R\),
\(t_0<t\in\mathbb R\)
Smaller to greater: =
Name: Newton-Leibniz formula

Name: A Riemann sum
Definition: \(\sum_{i=1}^n v(\tau_i)\Delta t_i\) for \(\tau_i\in[t_{i-1},t_i]\) and \((t_i)\) increasing
Page: 332
Chapter: 6
Section: 1.1
Variables: \(s:\mathbb R\to\mathbb R\),
\(v=s'\)

Name: A partition of \([a,b]\)
Definition: A finite increasing sequence \(a=x_0<\dots<x_n=b\)
Page: 333
Chapter: 6
Section: 1.2 a
Variables: \(a<b\in\mathbb R\)
Denoted by: \(P\)

Name: An interval of \(P\)
Definition: An interval \([x_i,x_{i+1}]\)
Page: 333
Chapter: 6
Section: 1.2 a
Variables: \(a<b\in\mathbb R\)
\(P=(x_0,\dots,x_n)\) a partition of \([a,b]\)

Name: The mesh of \(P\)
Definition: The volume (length) of the greatest interval of \(P\)
Page: 333, 110
Chapter: 6, 11
Section: 1.2 a, 1.1b
Variables: \(a<b\in\mathbb R^n\)
\(P\) a partition of \(I_{a,b}\)
Notation: \(\lambda(P)\)
Index: 1, 4
Kind: Def, Def

Name: Distinguished points of \(P\)
Definition: A function from \(P_i\in P\) to some element of \(P_i\)
Page: 333, 110
Chapter: 6, 11
Section: 1.2 a, 1.1b
Variables: \(a<b\in\mathbb R\)
\(P=(x_0,\dots,x_n)\) a partition of \([a,b]\)
Index: 2, 5
Kind: Def, Def
Denoted by: \(\xi\)

Name: The set of partitions with distinguished bases
Page: 333
Chapter: 6
Section: 1.2 b
Variables: \(a<b\in\mathbb R\)
Notation: \(\mathcal P\)
Kind: Def

Definition: All partitions with distingished points \((P,\xi)\) on \([a,b]\) for which \(\lambda(P)<d\)
Page: 333
Chapter: 6
Section: 1.2 b
Variables: \(a<b\in\mathbb R\)
\(d>0\)
Context: \(\mathcal P\)
Notation: \(B_d\)
Kind: Def

Name: The base of \(\mathcal P\)
Definition: \(\{B_d\mid d>0\}\) with \(B_d\) the set partitions with distinguished points and mesh at most \(d
Page: 333
Chapter: 6
Section: 1.2 b
Variables: \(a<b\in\mathbb R\)
Context: \(\mathcal P\) the set of partitions with distinguished points
Notation: \(\mathcal B\)
Kind: Def

Name: The Riemann sum of \(f\) corresponding to \((P,\xi)\)
Definition: [$]\sum_{P_i\in P}f(\xi_i)\abs{P_i}[/$]
Page: 333,110
Chapter: 6, 11
Section: 1.2 c,1.1 c
Variables: \(\mathbb F\) a real vector space
\(I\) a parallelepiped,
\(f:I\to\mathbb F\)
\((P,\xi)\) a partition of \(I\) with distinguished elements
Notation: \(\sigma(f;P,\xi)\)
Index: 3, 6
Kind: Def, Def

Definition: [$]\defFun{I\to\mathbb F}{\mathbb F}{f}{\lim_{P\in\mathcal B}\sigma(f;P)}[/$]
Page: 334, 110, 349
Section: 1.2 d, 1.2 c, 2.1
Chapter: 6, 11, 6
Variables: \(\mathbb F\) a real vector space
\(I\) a parallelepiped,
Notation: \(\int_I\)
Name: The Riemann integral operator on \(I\)
Index: Def, Def, Theo
Kind: 4, 7, 1
Linear: Linear functional

Definition: \(\lim_{P\in\mathcal B}\sigma(f;P)\)
Page: 334, 352, 110
Section: 1.2 d, 2.3a 1.2 c,
Chapter: 6, 6, 11
Variables: \(\mathbb F\) a real vector space
\(I\) a parallelepiped,
\(f:I\to\mathbb F\)
Notation: [$]\int_I f(x)\dd x[/$]
Name: The Riemann integral of \(f\) on \(I\)
Index: 4, 3, 7
Kind: Def, theo, Def
Monotonic: Yes
Assuming: This value is correctly defined

Definition: \(a\)
Page: 334
Chapter: 6
Section: 1.2 d
Variables: \(S\) the lexical symbol [$]\int_a^b f(x)\dd x[/$]
Index: 4
Kind: Def
Denoted by: The lower limit of integration of \(S\)

Name: The upper limit of integration of [$]\int_a^b f(x)\dd x[/$]
Definition: \(b\)
Page: 334
Chapter: 6
Section: 1.2 d
Context: [$]\int_a^b f(x)\dd x[/$]
Index: 4
Kind: Def

Name: The integrand of \(S\)
Definition: \(f\)
Page: 334
Chapter: 6
Section: 1.2 d
Variables: \(S\) the lexical symbol [$]\int_a^b f(x)\dd x[/$]
Index: 4
Kind: Def

Name: The differential form of \(S\)
Definition: [$]f(x)\dd x[/$]
Page: 334
Chapter: 6
Section: 1.2 d
Variables: \(S\) the lexical symbol [$]\int_a^b f(x)\dd x[/$]
Index: 4
Kind: Def

Name: The variable of integration of \(S\)
Definition: \(x\)
Page: 334
Chapter: 6
Section: 1.2 d
Variables: \(S\) the lexical symbol [$]\int_a^b f(x)\dd x[/$]
Index: 4
Kind: Def

Name: A Riemann integrable function
Definition: whose Riemann integral is defined.
Page: 335, 339, 111, 337,
Chapter: 6, 11, 6
Section: 1.2 d, 1.1d, 1.3b
Variables: \(\mathbb F\) a field
Index: 5, 3, 2
Kind: Def, Cor, Def, Prop
Definition2: \(\lim_{\lambda(P)\to0}\sum_{i=1}^n\omega(f,\delta_i)\Delta x_i=0\) where the limits is taken over partitions of \([a,b]\)
Example: Monotonic unary functions
Conjdef: A function \([a,b]\to\mathbb F\) such that, equivalently
Example2: Riemann function \(\mathbb R\to\mathbb R; x\mapsto \left\{\begin{array}{ll}\frac 1n,&\text{if }x=\frac{m}{n}\text{ is in lowest terms}\in \mathbb Q\\0,&\text{if }x\in\mathbb R\setminus\mathbb Q\end{array}\right.\)

Smaller0: \(f\) is Riemann integrable on \(I\)
Greater0: \(f\) is bounded on \(I\)
Page: 335, 111
Chapter: 6, 11
Section: 1.3 a, 1.2d
Variables: \(V\) a real vector space
Let \(I\) a parallelepiped,
\(f:I\to V\)
Index: , 1
Kind: Prop, Prop
Counterexample: Characteristic function of \(\mathbb Q\)

Name: A refinement of \(P\)
Definition: A partition of \([a,b]\) which is a superset of \(P\)
Page: 336
Chapter: 6
Section: 1.3 b
Variables: \(a<b\in\mathbb R\)
\(P=(x_i)\) a partition of \(P\)
Context: Integration theory
Denoted by: \(\tilde P\)

Definition: The \(j\)-th element of \([x_{i-1},x_i]\) in \(\tilde P\)
Page: 336
Chapter: 6
Section: 1.3 b
Variables: \(a<b\in\mathbb R\)
\(P=(x_i)\) a partition of \(P\)
\(\tilde P\) a refinement of \(P\)
Notation: \(x_{ij}\)

Definition: The interval \([x_{i-1},x_i]\)
Page: 336
Chapter: 6
Section: 1.3 b
Variables: \(a<b\in\mathbb R\)
\(P=(x_i)\) a partition of \(P\)
Notation: \(\Delta_i\)
Index: 2
Kind: Prop
Extra: the size of the interval \(\Delta x_{i}\)

Definition: \(x_i-x_{i-1}\)
Page: 336
Chapter: 6
Section: 1.3 b
Variables: \(a<b\in\mathbb R\)
\(P=(x_i)\) a partition of \(P\)
Notation: \(\Delta{x_i}\)
Index: 2
Kind: Prop
Definition2: the length of \(\Delta_i\)

Smaller0: \(\lim_{P\in\mathcal B}\sum_{i=1}^n\omega(f;\Delta_i)\Delta x_{i}=0\)
Greater0: \(f\) is continuous almost everywhere
Page: 337-339, 341, 342
Chapter: 6
Section: 1.3 b, c
Variables: a \(\mathbb R\)-vector space
\(a\lt b\)
\(f:[a,b]\to X\)
Index: 2, 2, 3, 2', 3
Kind: Prop, Cor, Prop, Prop
Smaller to greater: iff
Greater1: \(f\) is bounded
Smallest: \(f\in\mathcal R[a,b]\)
Greatest: \(\underline I=\overline I\)

Name: The set of Riemann integrable functions defined on \(E\)
Page: 335, 342, 124, 349
Chapter: 6, 11, 6
Section: 1.2 d, 1.3c, 3.1, 2.1
Variables: \(X\) a \(\mathbb R\)-vector space
\(E\subseteq\mathbb R^n\)
Context: Analysis
Notation: \(\mathcal R(E)\)
Index: 5, 4, 1, 1
Kind: Def, Prop, Prop, theo
Typ: vector space
Assuming: \(E\) bounded

Definition: \(\text{inf}_{x\in I_i}f(x)\)
Page: 340
Chapter: 6,11
Section: 1,1.3 a
Variables: \(n\in\mathbb N\),
\(I\subseteq\mathbb R^n\) an interval
\(f:I\to\mathbb R\)
\(P=\{I_i\}\) a partition of \(I\)
Notation: \(m_i\)

Definition: \(\text{sup}_{x\in I_i}f(x)\)
Page: 340,116
Chapter: 6,11
Section: 1,1.3 a
Variables: \(n\in\mathbb N\),
\(I\subseteq\mathbb R^n\) an interval
\(f:I\to\mathbb R\)
\(P=\{I_i\}\) a partition of \(I\)
Notation: \(M_i\)

Name: The upper Riemann sum of \(f\) on \([a,b]\) corresponding to \(P\)
Definition: \(\sum_{i=1}^nM_i\Delta x_i\)
Page: 340,116
Chapter: 6,11
Section: 1,1.3 a
Variables: \(n\in\mathbb N\),
\(I\subseteq\mathbb R^n\) an interval
\(f:I\to\mathbb R\)
\(P=\{I_i\}\) a partition of \(I\)
Notation: \(S(f;P)\)
Index: 1
Kind: Lem,
Definition2: \(\text{sup}_{\xi}\sigma(f;P,\xi)\)
Name2: The upper Darboux sum of \(f\) on \(I\) corresponding to \(P\)

Name: The lower Riemann sum of \(f\) on \([a,b]\) corresponding to \(P\)
Definition: \(\sum_{i=1}^nm_i\Delta x_i\)
Page: 340, 116
Chapter: 6,11
Section: 1,1.3 a
Variables: \(n\in\mathbb N\),
\(I\subseteq\mathbb R^n\) an interval
\(f:I\to\mathbb R\)
\(P=\{I_i\}\) a partition of \(I\)
Notation: \(s(f;P)\)
Index: 1
Kind: Lem
Definition2: \(\text{inf}_{\xi}\sigma(f;P,\xi)\)
Name2: The lower Darboux sum of \(f\) on \(I\) corresponding to \(P\)

Name: A \(n\)-dimensional set of measure zero in the sense of Lebesgue
Definition: for each \(\epsilon>0\),
Page: 344, 112
Chapter: 6, 11
Section: 1.3 d, 1.2a
Context: \(\mathbb R^n\)
Index: 7, 9, 1, 2
Kind: Def, Def, Ex, ex
Definition2: there exists a covering of \(E\) by a countable union \(\bigcup_{n}I_n\) of intervals such that
Example: \(\mathbb Q^n\)
Conjdef: A set \(E\subseteq\mathbb R^n\) such that
Definition3: [$]\sum\abs{I_n}<\epsilon[/$]
Example2: A set of \((x,f(x))\) with \(f:I\to\mathbb R\) for \(I\subseteq \mathbb R^{n-1}\).
Hide remaining def: yes

Name: The sets of reals of measure zero in the sens of Lebesgue
Page: 344, 112
Section: 1.3 d, 1.2a
Chapter: 6, 11
Index: 2, 2
Kind: Lem, Lem,
Extra: measure of infinite product is not clear
Under: subset
Under2: union
Condition2: countable
Under3: product
Condition3: finite
Closure4: contains
Under4: interval
Condition4: degenerate
Under5: set
Condition5: finite cardinality

Name: A property which holds almost everywhere
Definition: A property which is true except on a set of measure zero.
Page: 344
Chapter: 6
Section: 1.3 d
Index: 8
Kind: Def

Question: Example of composition on integrable functions, not integrable
Answer: [$]\abs{\text{sgn}}[/$]
Page: 347
Chapter: 6
Section: 1.3 c

Question: [$]\int_a^bf(x)\dd x+\int_b^cf(x)\dd x[/$]
Answer: equals [$]\int_a^cf(x)\dd x[/$]
Page: 350
Chapter: 6
Section: 2.2
Variables: \(\mathbb F\) a real vector space
\(a<b<c\)
\(f\in\mathcal R[a,c]\)

Definition: [$]-\int_a^bf(x)\dd x[/$]
Page: 350
Chapter: 6
Section: 2.2
Variables: \(\mathbb F\) a real vector space
\(a<b\)
\(f\in\mathcal R[a,b]\)
Context: Riemann Integration
Notation: [$]\int_b^af(x)\dd x[/$]

Question: [$]\int_a^af(x)\dd x[/$]
Answer: 0
Page: 350
Chapter: 6
Section: 2.2
Variables: \(\mathbb F\) a real vector space
\(a<b\)
\(f\in\mathcal R[a,c]\)
Context: Riemann Integration

Name: An additive (oriented) interval function
Definition: A function \(f:[a,b]^2\to\mathbb F\) such that, for each \(a\le\alpha\le\beta\le\gamma\le b\), \(I(\alpha,\beta)+I(\beta,\gamma)=I(\alpha,\gamma)\)
Page: 351
Chapter: 6
Section: 2.2
Variables: \(\mathbb F\) a group
\(a<b\)
Index: 1
Kind: Def
Example: the integral of an integrable function, for \([a,b]\) being the domain where it is integrable
Denoted by: \(I\)

Smaller0: [$]\abs{\int_Ef(x)\dd x}[/$]
Greater0: [$]\int_E\abs{f(x)}\dd x[/$]
Page: 352
Chapter: 6
Section: 2.3a
Variables: \(n\in\mathbb N\)
\(V\) a complete normed space
\(f\in\mathcal R(E,V)\)
\(E\subseteq \mathbb R^n\)
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)

Smaller0: \(m\le f(x)\le M\)
Greater0: [$]\in [m\abs{b-a},M\abs{b-a}][/$]
Page: 353
Chapter: 6
Section: 2.3b
Variables: \(\mathbb F\) a field
\(f\in\mathcal R[a,b]\)
\(m,M\in\mathbb R\)
Index: 1
Kind: Cor
Prefix Greater: [$]\int_a^b\abs{f(x)}\dd x[/$]

Smaller0: [$]\abs{f}(x)\le C[/$]
Greater0: [$]\le C\abs{b-a}[/$]
Page: 352
Chapter: 6
Section: 2.3a
Variables: \(\mathbb F\) a field
\(f\in\mathcal R[a,b]\)
\(C\in\mathbb R\)
Index: 3
Kind: Theo
Prefix Greater: [$]\int_a^b\abs{f(x)}\dd x[/$]

Question: [$]\frac{\int_a^b f(x)\dd x}{b-a}[/$]
Answer: is of the form \(\mu\)for \(\inf f(x)\le\mu\le\sup f(x)\)
Page: 353
Chapter: 6
Section: 2.3b
Variables: \(\mathbb F\) a field
\(f\in\mathcal R[a,b]\)
Index: 2
Kind: Cor

Smaller0: \(g\) is nonnegative or nonpositive
Greater0: is of the form [$]\mu\int_a^bg(x)\dd x[/$] for some \(\inf f(x)\le\mu\le\sup f(x)\)
Page: 352
Chapter: 6
Section: 2.3b
Variables: \(\mathbb F\) a field
\(f,g\in\mathcal R[a,b]\)
Index: 3
Kind: Theo
Prefix Greater: [$]\int_a^b(f(x)\cdot g(x))\dd x[/$]
Name: First mean-value theorem for the integral

Smaller0: \(g\) is nonnegative or nonpositive
Greater0: is of the form [$]f(\xi)\int_a^bg(x)\dd x[/$] for some \(a\le\xi\le b\)
Page: 353
Chapter: 6
Section: 2.3b
Variables: \(\mathbb F\) a field
\(f,g\in\mathcal R[a,b]\)
Index: 5
Kind: Theo
Prefix Greater: [$]\int_a^b(f(x)\cdot g(x))\dd x[/$]
Smaller1: \(f\in\mathcal C[a,b]\)

Smaller0: \(g\) is monotonic
Greater0: is of the form [$]g(a)\int_a^\xi f(x)\dd x+g(b)\int_\xi^bf(x)\dd x[/$] for \(\xi\in[a,b]\)
Page: 358
Chapter: 6
Section: 2.3b
Variables: \(\mathbb F\) a field
\(f,g\in\mathcal R[a,b]\),
Kind: Theo
Prefix Greater: [$]\int_a^b(f(x)g(x))\dd x[/$]
Name: Second mean-value theorem for the integral
Name2: Bonnet's formula

Smaller0: \(g\) is nonnegative nonincreasing
Greater0: is of the form [$]g(a)\int_a^\xi f(x)\dd x[/$] for \(\xi\in[a,b]\)
Page: 356
Chapter: 6
Section: 2.3b
Variables: \(\mathbb F\) a field
\(f,g\in\mathcal R[a,b]\)
Index: 4
Kind: Lem
Prefix Greater: [$]\int_a^b(f(x)g(x))\dd x[/$]

Smaller0: \(\sum_{i=1}^na_ib_i\)
Greater0: \(\sum_{i=1}^{n}\left(\sum_{j=1}^ia_i\right)(b_i-b_{i+1})+\left(\sum_{j=1}^na_i\right)b_n\)
Page: 355
Chapter: 6
Section: 2.3c
Variables: \(\mathbb F\) a ring
\(a_1,b_1,\dots,a_n,b_n\in\mathbb F\)
\(b_{n+1}=0\)
Smaller to greater: =
Name: Abel’s transformation
Construction: [latex]\begin{eqnarray*}&\sum_{i=1}^na_ib_i\\
=&\sum_{i=1}^n(\sum_{j=1}^ia_j-\sum_{j=1}^{i-1}a_j)b_i\\
=&\sum_{i=1}^n\sum_{j=1}^ia_jb_i-\sum_{i=1}^n\sum_{j=1}^{i-1}a_jb_i\\
=&\sum_{i=1}^n\sum_{j=1}^ia_jb_i-\sum_{i=0}^{n-1}\sum_{j=1}^{i}a_jb_{i+1}\\
=&\sum_{j=1}^na_jb_n+\sum_{i=1}^{n-1}\sum_{j=1}^ia_i(b_j-b_{j+1})-\sum_{j=1}^0b_1\\
=&\sum_{j=1}^na_jb_n+\sum_{i=1}^{n-1}\sum_{j=1}^ia_i(b_j-b_{j+1})
\end{eqnarray*}[/latex]

Smaller0: \((b_i)\) nonincreasing
Greater0: belongs to \([mb_1,Mb_1]\)
Page: 355
Chapter: 6
Section: 2.3c
Variables: \(\mathbb F\) a ring
\(a_1,b_1,\dots,a_n,b_n\in\mathbb F\)
Prefix Greater: \(\sum a_ib_i\)
Smaller1: \(m\le\sum_{j=1}^ia_j\le M\) for each \(0\le i\le n\)
Construction: Using Abel's transformation
\(\sum_{i=1}^na_ib_i=\sum_{i=1}^{n}\left(\sum_{j=1}^ia_i\right)(b_i-b_{i+1})\in [mb_n+\sum_{i=1}^{n-1}m(b_i-b_{i+1}),Mb_n+\sum_{i=1}^{n-1}M(b_i-b_{i+1})]=[mb_1,Mb_1]\)
Smaller2: \(b_n\ge0\)

Definition: [$]\defFun{[a,b]}{\mathbb F}{x}{\int_a^xf(t)\dd t}[/$]
Page: 356
Section: 2.3c
Chapter: 6
Variables: \(\mathbb F\) a ring
\(f\in\mathcal R[a,b]\)
Notation: \(F\)
Index: Lem
Kind: 3
Continuous: continuous