Title: Theory / Unimportant / Math / Analysis / Real Pugh/{ / 5 - Multivariable
Tags: fromMath mathToDef automaticImplies hasSeenConjDef Wrong_taylor_series list mathToSee mid1516974248815 toUnary typToBinary
On every note of this deck, unless a note says otherwise:
Chapter: 5

Name: The operator norm
Definition: \(\text{sup}\left\{\frac{|Tv|_W}{|v|_V}:v\ne0 \right\}\).
Page: 279
Chapter: 5
Section: 1
Variables: \(V,W\) be normed spaces and
\(T\in\mathcal L(V,W)\).
Notation: \(\parallel T\parallel\)

Smaller0: [$]\norm{T}<\infty[/$]
Greater0: \(T\) is continuous
Page: 279
Section: 1
Variables: \(V,W\) be normed spaces
\(T\in\mathcal L(V,W)\).
Index: 2
Kind: Theo
Smaller to greater: iff
Smallest: \(T\) is uniformly continuous
Greatest: \(T\) is continuous at the origin

Name: A function differentiable at \(p\).
Definition: A function which admits a derivative at \(p\).
Page: 282
Chapter: 5
Section: 2
Variables: \(U\) be an open subset of \(\mathbb R^n\),
\(p\in U\).

Name: The \(i\)-th derivative of \(f\) at point \(p\)
Page: 291
Chapter: 5
Section: 3
Variables: \(U\) be an open subset of \(\mathbb R^n\),
\(i\in\mathbb N\) and
\(f:U\to\mathbb R^m\).
Notation: \(D^if\)
Typ: \(U\to\mathcal L((\mathbb R^n)^i,\mathbb R^m)\)

Name: A uniformly \(C^r\) convergent sequence of \(C^r\) function
Definition: A sequence \((f_k):U\to\mathbb R^m\) such that for \(i\) from 0 to \(r\), \(D^if_k\rightrightarrows D^if\) for some \(f:U\to\mathbb R^m\).
Page: 295
Chapter: 5
Section: 3
Variables: \(U\) be an open subset of \(\mathbb R^n\).

Name: An uniformly \(C^r\) Cauchy sequence
Definition: A sequence \((f_k):U\to\mathbb R^m\) such that, for each \(\epsilon>0\) there is an \(N\) such that for all \(k,l\ge N\), for all \(0\le i\le r\) and all \(x\in U\) we have \(\parallel (D^if_k)_x-(D^if_l)_x\parallel<\epsilon.\)
Page: 295
Chapter: 5
Section: 3
Variables: \(U\) be an open subset of \(\mathbb R^n\).

Name: The \(C^r\) norm of \(f\)
Definition: [$]\text{sup}_{x\in U, 0\le i\le r}\{\norm{ (D^if)_x}\}[/$].
Page: 296
Chapter: 5
Section: 3
Variables: \(U\) be an open subset of \(\mathbb R^n\)
\(f:U\to\mathbb R^m\)
Notation: [$]\norm{f}_r[/$]

Definition: The set of functions \(f\) such that [$]\norm{f} < \infty[/$].
Page: 296
Chapter: 5
Section: 3
Variables: \(U\) be an open subset of \(\mathbb R^n\).
Notation: \(C^r(U,\mathbb R^m)\)

Name: A Banach space
Definition: complete
Page: 296
Chapter: 5
Section: 3
Index: 20
Kind: Cor
Definition2: normed
Conjdef: A vector space

Name: The \(f(x_0,y_0)\)-locus of \(f\)
Definition: The set of pairs \((x,y)\in U\) such that \(f(x,y)=f(x_0,y_0)\).
Page: 297
Chapter: 5
Section: 4
Variables: \(U\) open subset of \(\mathbb R^n\times\mathbb R^m\)
\(f:U\to\mathbb R^m\)
\((x_0,y_0)\in U\).

Name: The implicit function defined by \(f(x,y)=f(x_0,y_0)\)
Definition: A function \(g:\mathbb R^n\to\mathbb R^m\) such that for all \(x\), \(f(x,g(x))=f(x_0,y_0)\).
Page: 297
Chapter: 5
Section: 4
Variables: \(U\) open subset of \(\mathbb R^n\times \mathbb R^m\)
\(f:U\to \mathbb R^m\).
\((x_0,y_0)\in U\).

Smaller0: \(f\in C^r\)
Greater0: exists near \((x_0,y_0)\)
Page: 298
Section: 4
Variables: \(U\) open subset of \(\mathbb R^n\times \mathbb R^m\)
\(f:U\to \mathbb R^m\)
\(1\le r\le\infty\)
\((x_0,y_0)\in U\)
Index: 22
Kind: Theorem
Hide middle: yes
Prefix Greater: the implicit function \(g\)
Name: Implicit function Theorem
Greater1: is \(C^r\)
Greater2: its graph is the \(f(x_0,y_0)\)-locus of \(f\)

Smaller0: \(f\) is of class \(C^{\prime1}\)
Greater0: \[f(q)-f(p)\]
Section: 2
Variables: \(f:U\to\mathbb R^m\).
Hide middle: yes
Smaller1: the segments \([p,q]\) is contained in \(U\)
Greater1: [$$]\int_0^1(Df)_{p+t(q-p)}\dd t(q-p)[/$$]
Connect Greater0: =

Definition: The set of \(m\times n\) matrices with real entries
Page: 277
Chapter: 5
Section: 1
Variables: \(m,n\in\mathbb N\)
Notation: \(\mathcal M(m,n)\)

Definition: The set of linear transformations from \(\mathbb R^n\) to \(\mathbb R^m\).
Page: 278
Chapter: 5
Section: 1
Variables: \(m,n\in\mathbb N\)
Notation: \(\mathcal L(\mathbb R^n,\mathbb R^m)\)

Smaller0: the rank of \(T\)
Greater0: the dimension of the range of \(T\)
Page: 301
Variables: \(n,m\in\mathbb N\)
\(T\in\mathcal L(\mathbb R^n\to\mathbb R^m)\).
Smaller to greater: is equal to
Hide middle: yes

Smaller0: \(T\) is injective
Greater0: \(n\)
Page: 302
Variables: \(n,m\in\mathbb N\)
\(T\in\mathcal L(\mathbb R^n\to\mathbb R^m)\).
Smaller to greater: iff
Prefix Greater: \(T\)'s rank is

Smaller0: \(T\) is surjective
Greater0: \(T\)'s rank is \(m\)
Page: 302
Variables: \(n,m\in\mathbb N\)
\(T\in\mathcal L(\mathbb R^n\to\mathbb R^m)\).
Smaller to greater: iff

Name: Equivalent functions
Definition: Two functions \(f:A\to B\), \(g:C\to D\) such that there exists bijections \(\alpha:A\to C\) and \(\beta:B\to D\) such that \(g=\beta\circ f\circ \alpha^{-1}\).
Page: 302
Chapter: 5

Name: \(C^r\)-equivalence
Definition: The relation over \(C^r\) functions \(f:A\to B\), \(g:C\to D\) such that there exists \(C^r\) bijections \(\alpha:A\to C\) and \(\beta:B\to D\) such that \(g=\beta\circ f\circ\alpha^{-1}\).
Page: 302
Chapter: 5
Notation: \(\equiv_r \)
Index: 24
Kind: Lem
Typ: Equivalence relation

Name: The linear projection
Definition: \((x_1,\dots,x_n)\mapsto(x_1,\dots,x_n,0,\dots,0)\).
Page: 303
Chapter: 5
Variables: \(0\le n\le m\) integers.
Context: \(\mathbb R^n\to\mathbb R^m\)
Notation: \(P\)

Smaller0: \(f\in C^r\) is constant-rank-k
Greater0: \(C^r\) equivalent to a linear projection onto a \(k\)-dimensional subspace
Page: 303
Variables: \(U\subseteq\mathbb R^n\)
\(f:U\to\mathbb R^m\).
Index: 25
Kind: Theo
Prefix Greater: \(f\) is
Name: Rank Theorem

Smaller0: \(f\in C^r\) has rank \(k\) at \(p\)
Greater0: locally \(C^r\) equivalent to a map of the form \(G(x,y)=(x,g(x,y))\) where \(g:\mathbb R^n\to\mathbb R^{m-k}\) is \(C^r\) and \(x\in\mathbb R^k\)
Page: 303
Variables: \(U\subseteq\mathbb R^n\)
\(f:U\to\mathbb R^m\).
Index: 26
Kind: Cor
Prefix Greater: \(f\) is

Smaller0: \(f\in C^r\) has rank \(1\) at \(p\)
Greater0: in a neighborhood of \(p\), the level sets \(\{x\in U:f(x)=c\}\) for a stack of \(C^r\) nonlinear discs of dimension \(n-1\)
Page: 303
Variables: \(U\subseteq\mathbb R^n\)
\(f:U\to\mathbb R\)
\(p\in U\).
Index: 27
Kind: Cor

Definition: The function which sends \(p\in U\) to a map \(T\) such that \(\lim_{|v|\to0} \frac{f(p+v)-f(p)-T(v)}{|v|}=0\)
Page: 282
Section: 2
Chapter: 5
Variables: \(U\) be an open subset of \(\mathbb R^n\)
\(f:U\to \mathbb R^m\).
Notation: \(Df\)
Name: The total derivative of \(f\)
Linear: \(U\to\mathcal L(\mathbb R^n,\mathbb R^m)\).
Name2: The Fréchet derivative of \(f\)

Smaller0: the derivative of its \(i\)th component
Greater0: the \(i\)th component of the derivative
Page: 288
Section: 2
Variables: \(U\subseteq\mathbb R^n\),
\(f:U\to \mathbb R^m\) differentiable and
\(p\in U\).
Index: 10
Kind: Theo
Smaller to greater: is equal to
Hide middle: yes

Name: The \(ij^{\text{th}}\) partial derivative of \(f\) at \(p\)
Definition: \(\lim_{t\to 0}\frac{f_i(p+te_j)-f_i(p)}{t}\).
Page: 284
Chapter: 5
Section: 2
Variables: \(U\) be an open subset of \(\mathbb R^n\),
\(p\in U\),
\(f:U\to \mathbb R^m\),
\(0<i\le m\)
\(0<j\le n\)
Notation: [$]\partialderivative{f_i(p)}{x_j}[/$]

Smaller0: the derivative of \(f\) is invertible
Greater0: \(f\) is a local diffeomorphism
Page: 301
Section: 4
Variables: \(U\) open subset of \(\mathbb R^n\times \mathbb R^m\)
\(f:U\to \mathbb R^m\).
Index: 23
Kind: Theorem
Name: Inverse function Theorem

Smaller0: \(\parallel T\circ S\parallel\)
Greater0: \(\parallel T\parallel \parallel S\parallel\)
Page: 249
Section: 1
Variables: \(U,V,W\) normed space,
\(S:U\to V, T:V\to W\) two linear maps.
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)

Question: Topologically \(T\) is
Answer: continuous.
Page: 278
Chapter: 5
Section: 1
Variables: \(W\) a vector space and \(T:\mathbb R^n\to W\).
Index: 3
Kind: Theorem

Smaller0: \(T\) is an isomorphism
Greater0: \(T\) is an homeomorphism
Page: 280
Section: 1
Variables: \(n\in\mathbb N\),
\(W\) a vector space
\(T\in\mathcal L(\mathbb R^n,W)\).
Index: 3
Kind: Theorem

Name: The conorm of \(T\)
Definition: The greatest \(c\) such that \(|Tv|\ge c|v|\).
Page: 281
Chapter: 5
Section: 1
Variables: \(V,W\) normed vector spaces and \(T:\mathcal L(V,W)\).

Smaller0: \(f\) is differentiable
Greater0: \(f\) is continuous
Page: 283
Section: 2
Variables: \(f:\mathbb R^n\to \mathbb R^m\).
Index: 6
Kind: Theorem
Hide middle: yes

Smaller0: the total derivative exists
Greater0: the partial derivative exists and they are the entries of the matrix that represents the total derivative
Page: 284
Section: 2
Variables: \(U\) be an open subset of \(\mathbb R^n\),
\(p\in U\),
\(f:U\to \mathbb R^m\)
Index: 7
Kind: Cor

Smaller0: the partial derivatives of \(f\) exist and are continuous
Greater0: \(f\) is differentiable
Page: 285
Section: 2
Variables: \(U\) be an open subset of \(\mathbb R^n\), \(p\in U\), let \(f:U\to \mathbb R^m\)
Index: 8
Kind: Theo
Smaller to greater: =
Hide middle: yes
Prefix Greater: "background-color: rgb(255, 255, 255)"> implies
Suffix Greater: .

Question: \(D(f+cg)=\)
Answer: \(Df+cDg\).
Page: 285
Chapter: 5
Section: 2
Variables: \(U\) be an open subset of \(\mathbb R^n\), let \(f,g:U\to\mathbb R^m\).
Index: 9a
Kind: Theorem

Smaller0: \(f\) is constant
Greater0: \(0\)
Page: 285
Section: 2
Variables: \(U\) be an open subset of \(\mathbb R^n\),
\(f:U\to \mathbb R^m\).
Index: 9b
Kind: Theo
Prefix Greater: \(D(f)=\)

Question: \(D(f(x))=\)
Answer: \(f\)
Page: 285
Chapter: 5
Section: 2
Variables: \(U,V\) be vector spaces, let \(f\in\mathcal L(U,V)\).
Index: 9b
Kind: Theorem

Smaller0: \(f,g\) are differentiable
Greater0: \(Dg\circ Df\)
Page: 285
Section: 2
Variables: \(U\subseteq\mathbb R^n, V\subseteq\mathbb R^m\),
\(f:U\to V\)
\(g:V\to\mathbb R^k\).
Index: 9c
Kind: Theorem
Hide middle: yes
Prefix Greater: \(D(f\circ g)=\)
Name: Chain Rule

Smaller0: the \(m\) components \(f_i\) are differentiable at \(p\)
Greater0: The function \(f\) is differentiable at \(p\)
Page: 288
Section: 2
Variables: \(U\subseteq\mathbb R^n\),
\(f:U\to \mathbb R^m\) and
\(p\in U\).
Index: 10
Kind: Theo
Hide middle: yes

Smaller0: \(f\) is differentiable on \(U\) and the segment \([p,q]\) is contained in \(U\)
Greater0: [$$]\abs{f(q)-f(p)}\le \text{sup}\{\norm{ (Df)_x} :x\in U\} \abs{q-p}.[/$$]
Page: 288
Section: 2
Variables: \(f:U\to \mathbb R^m\),
\(p,q\in U\).
Index: 11
Kind: Theo
Hide middle: yes

Smaller0: \(f\) is differentiable and \((Df)_x=0\)
Greater0: \(f\) is constant
Page: 290
Section: 2
Variables: \(U\subseteq\mathbb R^n\)
\(f:U\to\mathbb R^m\).
Index: 13
Kind: Corollary
Hide middle: yes
Smaller1: \(U\) connected

Smaller0: \(f\) is continuous, and if [$]\partialderivative{f(x,y)}{y}[/$] exists and is continuous
Greater0: [$]\int_a^b f(x,y)\dd x[/$] of class \(C^1\)
Page: 290
Section: 2
Variables: \(a,b,c,d\in\mathbb R\), \(f:[a,b]\times(c,d)\to\mathbb R^m\).
Index: 14
Kind: Theo
Hide middle: yes
Suffix Smaller: ,

Question: Property of \((D^if)_p\)
Answer: It is linear in each of its \(i\)-th component.
Chapter: 5
Section: 3
Variables: \(U\) be an open subset of \(\mathbb R^n\) and
\(f:U\to \mathbb R^m\).

Smaller0: \((D^2f)_p\) exists
Greater0: [$](D^2f_k)_p(e_i,e_j)=\partialderivative{^ 2f_k(p)}{x_ix_j} [/$]
Page: 291
Section: 3
Variables: \(U\) be an open subset of \(\mathbb R^n\), \(p\in U\) and \(0<i\le n\), \(0<j\le m\), \(f:U\to\mathbb R^m\).
Index: 15
Kind: Theo

Question: Property of \((D^2f)_p\) if it exists.
Answer: \((D^2f)_p(v,w)=(D^2f)_p(w,v)\).
Page: 292
Chapter: 5
Section: 3
Variables: \(U\) be an open subset of \(\mathbb R^n\), \(p\in U\) and \(f:U\to\mathbb R^m\).
Index: 16
Kind: Theorem

Question: \((D^rf)_p(v_{\sigma 1},\dots,v_{\sigma r})=\)
Answer: \((D^rf)_p(v_1,\dots,v_r)\).
Page: 294
Chapter: 5
Section: 3
Variables: \(U\) be an open subset of \(\mathbb R^n\), \(p\in U\) and \(f:U\to\mathbb R^m\), let \(r\in\mathbb N\), \(v_1,\dots,v_r\in\mathbb R^n\) and \(\sigma\) a permutation of \(\{1,\dots,r\}\).
Index: 18
Kind: Corollary

Smaller0: \(f\) is second-differentiable
Greater0: [$]\partialderivative[2]{f_k(p)}{x_jx_i}[/$].
Page: 294
Section: 3
Variables: \(f:U\to\mathbb R^n\).
Index: 17
Kind: Cor
Hide middle: yes
Prefix Greater: [$]\partialderivative[2]{f_k(p)}{x_ix_j}[/$] is equal to

Smaller0: \((f_k)\) is uniform \(C^r\) convergent
Greater0: \((f_k)\) is Cauchy \(C^r\) convergent.
Page: 296
Section: 3
Variables: \(U\) be an open subset of \(\mathbb R^n\). Let \((f_k):U\to\mathbb R^m\).
Index: 19
Kind: Theorem
Smaller to greater: iff
Hide middle: yes

Name: A sublinear function on \(U\) to \(\mathbb R^m\)
Definition: \(f:U\to \mathbb R^m\) such that \(\lim_{v\to 0} \frac{f(v)}{|v|}=0\).
Page: 282
Chapter: 5
Section: 2
Variables: \(U\subseteq R^n\) open containing 0 and let .

Smaller0: \(f,g\) are differentiables
Greater0: \(Df\cdot g+f\cdot Dg\)
Page: 285
Section: 2
Variables: \(U\subseteq\mathbb R^n\) and
\(f,g:U\to\mathbb R^m\).
Index: 9d
Kind: Theorem
Hide middle: yes
Prefix Greater: \(D(f\cdot g)=\)
Name: Leibniz Rule

Question: Property of \(C^r(U,\mathbb R^m)\) with norm \(\parallel\ \parallel_r\).
Answer: It is a Banach space.
Page: 296
Chapter: 5
Section: 3
Variables: \(U\subseteq\mathbb R^n\) open
Index: 20
Kind: Corollary

Question: Term-by-term differentiation order \(\le r\) is
Answer: valid, i.e., for all \(s\le r\) we have \(D^sf=\sum_k D^sf_k\).
Page: 297
Chapter: 5
Section: 3
Variables: \((M_k)\in\mathbb R\) and
\((f_k):U\to\mathbb R^m\).
Index: 21
Name: \(C^r\) \(M\)-test

Smaller0: \(\sum M_k\) is a convergent series of constants and if \(\parallel f_k\parallel_r\le M_k\) for all \(k\)
Greater0: the series of functions \(\sum f_k\) converges in \(C^r(U,\mathbb R^m)\) to a function \(f\)
Page: 297
Section: 3
Variables: \((M_k)\in\mathbb R\) and
\((f_k):U\to\mathbb R^m\).
Index: 21
Hide middle: yes
Name: \(C^r\) \(M\)-test

Q: rank is:
-[...] by [...]
A: preserved; equivalence of functions
Page: 302
Chapter: 5
Index: 24
Kind: Lem