Title: Theory / Important / Math / Analysis / Zorich{ / 05, 08 Differentiation / 8 multiple variable
Tags: hasSeenConjDef TypToQuestion

Name: A differentiable functions at the point \(x\)
Definition: \(f(x+h)-f(x)\) is of the form \(L(x)h+\alpha(x;h)\) with \(L(x):X\to Y\) linear in \(h\) and \(\alpha(x;h)=_{\lim h\to0,x+h\in E}o(h)\)
Page: 432, 434, 456
Chapter: 8
Section: 2.1, 2.2, 4.2
Variables: \(X\subseteq\mathbb R^n,Y\subseteq\mathbb R^m\) normed vector spaces,
\(E\subseteq X\)
\(x\in E\)
Index: 1
Kind: Prop, theo
Definition2: Or, equivalently: \(f^i=\pi_i\circ f\) is differentiable, assuming \(Y=\mathbb R^n\), and \(\pi_i\) is the \(i\)-th projection function
Conjdef: such that
Definition3: there exists a neighborhood \(U\) of \(x\) such that
\((\frac{\partial f}{\partial x^i})_{1\le i\le m}\) exists on \(U\) and is continuous at \(x\), assuming \(Y=\mathbb R\)
DefType: A function \(f:E\to Y\)

Name: The differential of \(f\) at \(x\)
Definition: A linear function \(L(x):X\to Y\) such that \[f(x+h)-f(x)=L(x)h+\alpha(x;h)\] with \(\alpha(x;h)=o(h)\) as \(h\to 0\), \(x+h\in E\)
Page: 432, 435, 436, 437
Chapter: 8
Section: 2.1, 2.2, 2.3
Variables: \(E\subseteq \mathbb R^m\)
\(x\in E\)
\(f:\mathbb R^m\to \mathbb R^n\) differentiable
Notation: [$]\dd f(x)[/$]
Index: , 2, 30, 3
Kind: , prop, Eq, Prop
Definition2: \((h^1,\dots,h^m)\mapsto\partial _i f(x) h^i\) (implicit sum on indices \(i\), assuming \(x\) in the interior of \(U\))
Definition3: \((h^1,\dots,h^m)\mapsto Jh\) with \(J\) the Jacobian of \(f\) at \(x\)
Name2: The tangent mapping of \(f\) at \(x\)
Typ: Linear function in \(T\mathbb R^m_x\to T\mathbb R^n_{f(x)}\)
Assuming: A function as in the first definition exists
Notation2: \(Df(x)\)
Name3: The derivative mapping of \(f\) at \(x\)
Notation3: \(f'(x)\)

Name: The tangent space to \(\mathbb R^m\) at \(x\)
Definition: The vector space centred around \(x\)
Page: 433
Chapter: 8
Section: 2.1
Variables: \(x\in\mathbb R^m\)
Notation: \(T_x\mathbb R^m\)
Notation2: \(T\mathbb R^m(x)\)
Notation3: \(T\mathbb R^m_x\)

Name: A smooth \(k\)-dimensional surface in \(\mathbb R^n\)
Definition: for each \(x_0\in S\),
Page: 164
Chapter: 12
Section: 1
Variables: \(k,n\in\mathbb K\)
Index: 3
Kind: Def
Definition2: there exists a neighborhood \(U(x_0)\subseteq \mathbb R^n\) and
Conjdef: A set \(S\subseteq\mathbb R^n\) such that
Definition3: there exists a diffeomorphism \(\psi:U(x_0)\to I^n\) with \(I^n\) the open cube
Definition4: under which \(U(x_0)\cap S\) maps into the cube \(I^k=\{(t_1,\dots,t_k,0,\dots,0)\mid 0\le t_1,\dots,t_k\le 1\}\)
Hide remaining def: y

Name: The partial derivative of the function \(f(x)\) the \(x\) with respect to the variable \(x^i\)
Definition: \(\lim_{h^i\to 0}\frac{f(x^1,\dots, x^{i-1},x^i+h^i,x^{i+1},\dots,x^m)-f(x^1,\dots,x^i,\dots,x^m)}{h^i}\)
Page: 434, 435
Chapter: 8
Section: 2
Variables: \(U\subseteq\mathbb R^m\),
\(f:U\to\mathbb R^n\)
\((x^1,\dots,x^m)\) a limit point of \(U\)
Notation: [$]\pdv[i]{f}{x}(x)[/$]
Index: 26, 2
Kind: Def, eq
Definition2: \(D_{e_i}f(x)\) with \(e_i\) the basis vector \((0,\dots,0,1,0,\dots,0)\)
Notation2: \(\partial_i f(x)\)
Notation3: \(D_if(x)\)
Notation4: \(f'_{x^i}(x)\)

Question: A formula with an index in subscript and superscript means
Answer: there is an implicit summation over an index
Page: 435
Chapter: 8
Section: 2

Name: The Jacobian of \(f\) at \(x\)
Definition: [$](\partial _if^j(x))_{1\le i\le m,1\le j\le n}=
\left(
\begin{array}{lll}
\pdv{f^1(x)}{x^1}&\dots&\pdv{f^1(x)}{x^m}\\
\vdots&&\vdots\\
\pdv{f^n(x)}{x^1}&\dots&\pdv{f^n(x)}{x^m}
\end{array}\right)
[/$]
Page: 437
Chapter: 8
Section: 2.3
Variables: \(U\subseteq\mathbb R^m\),
\(f:U\to\mathbb R^n\)
\((x^1,\dots,x^m)\) an interior point of \(U\)
Notation: \(f'(x)\)
Name2: The Jacobi matrix of \(x\)

Name: The differentiation operator for \(U\to\mathbb R^m\) at \(x\)
Page: 439
Chapter: 8
Section: 3.1
Variables: \(U\subseteq\mathbb R^m\),
\(x\in U\)
Index: 1
Kind: Theo
Typ: Element of \(L(U\to\mathbb R^n, \mathbb R^m\to\mathbb R^n)\)

Name: The derivative of \(f\) at the point \(x\) with respect to the vector \(v\)
Definition: \(\lim_{t\to0}\frac{f(x+vt)-f(x)}{t}\)
Page: 444, 445, 446
Chapter: 8
Section: 3.2
Variables: \(E\subseteq\mathbb R^n\),
\(x\) a limit point of \(E\),
\(f:E\to\mathbb R^m\) differentiable at \(x\)
\(v\in T\mathbb Rm_{x}\)
Notation: \(D_v f(x)\)
Index: 1
Kind: Def
Definition2: [$](\dd f(x))v[/$], assuming [$]\dd f[/$] is defined at \(x\)
Definition3: [$]\abs{\mathrm{grad}f(x)}\cos\phi[/$], assuming \(v\) is unitary, for \(\phi\) the angle between the vectors \(v\) and [$]\mathrm{grad}f(x)[/$]
Name2: the derivative along the vector \(v\) at the point \(x\)
Definition4: [$]\langle\mathrm{grad}f(x),v\rangle[/$]
Definition5: [$]\sum\pdv{f}{x^i}(x)v^i[/$] where \(\{x^i\}\) is a base of \(\mathbb R^m\)

Question: \(D_{\lambda_1v_1+\lambda_2v_2}f(x)\)
Answer: equals \(\lambda_1D_{v_1}f(x)+\lambda_2D_{v_2}f(x)\)
Page: 445
Chapter: 8
Section: 3
Variables: \(E\subseteq\mathbb R^n\),
\(x\) a limit point of \(E\),
\(f:E\to\mathbb R^m\) differentiable at \(x\)
\(v_1,v_2\in T\mathbb R_x^m\)
\(\lambda_1,\lambda_2\in\mathbb R\)

Name: The gradiant of \(f\) at \(x\)
Definition: \(\xi\in\mathbb R^m\) such that for each \(v\in\mathbb R^m\), [$]\dd f(x)v=\langle \xi ,v\rangle[/$]
Page: 445
Chapter: 8
Section: 3
Variables: \(E\subseteq\mathbb R^n\),
\(x\) a limit point of \(E\),
\(f:E\to\mathbb R^m\) differentiable at \(x\)
\(v\in T\mathbb R_x^m\)
Index: 2
Kind: Def
Definition2: [$]\left(\pdv{f}{x^1},\dots,\pdv {f}{x^m}\right)(x_0)[/$] assuming a Cartesian coordinate of \(\mathbb R^m\)
Definition3: \(D_{e} f(x)e\), \(e\in\mathbb R^n\) unit vector, such that
that \(D_{e} f(x)\) is maximal,
Abbreviation: [$]\mathrm{grad}f(x)[/$]

Name: The directional derivative of \(f\) in the direction of \(v\) at \(x\)
Definition: [$]D_{\frac{v}{\norm{v}}}f(x)[/$]
Page: 446
Chapter: 8
Section: 3
Variables: \(E\subseteq\mathbb R^n\),
\(x\) a limit point of \(E\),
\(f:E\to\mathbb R^m\) differentiable at \(x\)
\(v\in T\mathbb R_x^m\)

Question: \(e\) can be written as
Answer: \((\cos\alpha_1,\dots,\cos\alpha_m)\) with \(\alpha_i\) the angle between \(e\) and \(e^i\)
Page: 446
Chapter: 8
Section: 3
Variables: \(e\in\mathbb R^m\)
Assuming: \(e\) unitary

Smaller0: \(f\) continuous and differentiable at \(x\)
Greater0: \((f^{-1})'\) exists and equal \((f')^{-1}\)
Page: 447
Chapter: 8
Section: 3.3
Variables: \(m\in\mathbb N\)
\(x\in\mathbb R^m\)
\(U\subseteq R^m\) neighborhood of \(x\)
\(f:U\to \mathbb R^m\)
Index: 4
Kind: Theo
Smaller1: \(f^{-1}\) exists and continuous on \(f(U)\)
Smaller2: \((f')^{-1}\) exists

Page: 447
Definition: Relation over paths \(y,z\) such that \(y(0)=z(0)\) and \(d(y(t),z(t))=o(t)\) as \(t\to 0\)
Chapter: 8
Section: 3.4
Variables: \(m\in\mathbb N\)
\(x_0\in\mathbb R^n\)
Name: Paths equivalent at \(x_0\)
Kind: Exo
Index: 1
Equivalence: yes

Name: A potential field
Definition: A vector field of the form [$]\mathtt{grad}U(x)[/$] for some \(U:G\mathbb R\)
Page: 449
Chapter: 8
Section: 3.4
Variables: \(m\in\mathbb N\)
\(G\subseteq\mathbb R^m\)
Index: 3
Kind: Exo
Intuition: A field that has a potential

Name: The potential of \(v\)
Definition: A function \(u:G\to\mathbb R\) such that [$]v=\mathtt{grad}u[/$]
Page: 449
Chapter: 8
Section: 3.4
Variables: \(m\in\mathbb N\)
\(G\subseteq\mathbb R^m\)
\(v:G\to \mathbb R\)
Index: 1
Kind: Exo
Extra: In physics, it is acutaly the inverse of the potential

Name: A streamline of \(v\)
Definition: A curve whose tangent at each point has the direction of \(v\)
Page: 450
Chapter: 8
Section: 3.4
Variables: \(m\in\mathbb N\)
\(G\subseteq\mathbb R^m\)
\(v:G\to \mathbb R\)
Index: 4
Kind: Exo

Name: \(f\) is homogeneous of degree \(n\)
Definition: \(f(\lambda x)=\lambda^nf(x)\) where the equality is defined
Page: 451
Chapter: 8
Section: 3
Variables: \(n,m\in\mathbb N\)
\(G\subseteq\mathbb R^m\)
\(f:G\to\mathbb R\)

Name: \(f\) is positive-homogeneous of degree \(n\)
Definition: \(f(\lambda x)=|\lambda|^nf(x)\) where the equality is defined
Page: 451
Chapter: 8
Section: 3
Variables: \(n,m\in\mathbb N\)
\(G\subseteq\mathbb R^m\)
\(f:G\to\mathbb R\)

Name: \(f\) is locally homogeneous of degree \(n\)
Definition: \(f\) is homogeneous of degree \(n\) in a neighborhood of each point of \(G\)
Page: 451
Chapter: 8
Section: 3
Variables: \(n,m\in\mathbb N\)
\(G\subseteq\mathbb R^m\)
\(f:G\to\mathbb R\)

Smaller0: \(f\) is locally homogeneous of degree \(n\)
Greater0: \(\sum_{i=1}^mx^i\frac{\partial f}{\partial x^i}(x^1,\dots,x^m)\)
Page: 451
Chapter: 8
Section: 3
Variables: \(n,m\in\mathbb N\)
\(G\subseteq\mathbb R^m\)
\(f:G\to\mathbb R\)
Index: 6
Kind: Ex
Smaller to greater: iff
Name: Euler identity for homogeneous function
Greater1: \(nf(x^1, \dots, x^m)\)
Connect Greater0: =

Name: The second partial derivative of \(f\) with respect to the variable \(x^i\) and \(x^j\)
Definition: \(\partial_j(\partial_i f)\)
Page: 457
Chapter: 8
Section: 4.3
Variables: \(m\in\mathbb N\)
\(G\subseteq\mathbb R^m\)
\(f:G\to\mathbb R\)
Notation: \(\partial_{ji}f(x)\)
Kind: Ex
Notation2: \(\frac{\partial^2f}{\partial x^j\partial x^i}(x)\)

Name: A partial derivative of order \(k\)
Definition: Some \(\partial_{i_1\dots i_k}f(x)\)
Page: 457
Chapter: 8
Section: 4.3
Variables: \(m\in\mathbb N\)
\(G\subseteq\mathbb R^m\)
\(f:G\to\mathbb R\)
Kind: Ex

Name: \(\partial_{i_1}(\partial_{i_2\dots i_k})(x)\)
Page: 458
Chapter: 8
Section: 4.3
Variables: \(m\in\mathbb N\)
\(G\subseteq\mathbb R^m\)
\(f:G\to\mathbb R\)
Notation: \(\partial_{i_1\dots i_k}f(x)\)
Kind: Ex
Notation2: \(\frac{\partial^kf}{\partial x^{i_1}\dots x^{i_k}}(x)\)

Smaller0: \(\partial_{ij}f(x)\) and \(\partial_{ji}f(x)\) exists and is continuous
Greater0: \(\partial_{ij}f(x)=\partial_{ji}f(x)\)
Page: 458
Chapter: 8
Section: 4.3
Variables: \(m\in\mathbb N\)
\(G\subseteq\mathbb R^m\)
\(f:G\to\mathbb R\)
\(1\le i,j\le m\)
Index: 3
Kind: Theo

Smaller0: \(\partial_if\) and \(\partial_jf\) exists in a neighborhood of \(x\)
Greater0: \(\partial_{ij}f(x)=\partial_{ji}f(x)\)
Page: 458, 474
Chapter: 8, 8
Section: 4.3, 4.7
Variables: \(m\in\mathbb N\)
\(G\subseteq\mathbb R^m\)
\(f:G\to\mathbb R\)
\(1\le i,j\le m\)
Index: 3, 2 a)
Kind: Theo, Exercice
Smaller1: \(\partial_{ij}f\) and exists and is continuous at \(x\)
Counterexample: [$$](x,y)\mapsto\left\{
\begin{array}{ll}
0,& \qif x=y=0\\
xy\frac{x^2-y^2}{x^2+y^2},\qelse
\end{array}
\right.[/$$]
\(\partial_{xy}f(0,0)=1\ne -1=\partial_{yx}f(0,0)\)

Definition: The set of functions all of whose partial derivatives up to order \(k\) inclusive are defined and continuous in \(G\)
Page: 459
Chapter: 8
Section: 4.3
Variables: \(m,k\in\mathbb N\)
\(G\subseteq\mathbb R^m\)
Notation: \(C^{(k)}(G;\mathbb R)\)
Notation2: \(C^{(k)}(G)\)

Smaller0: \(f\in C^{(k)}(G;\mathbb R)\)
Greater0: entirely determined by the multiset \(\{i_1\dots i_k\}\)
Page: 459
Chapter: 8
Section: 4.3
Variables: \(m,k\in\mathbb N\)
\(G\subseteq\mathbb R^m\)
\(f: G\to \mathbb R\)
Prefix Greater: \(\partial_{i_1\dots i_k}f(x)\) is

Smaller0: \(\phi^{(d)}(t)\)
Greater0: \(\left(\sum_{i}h^i\partial_i\right)^df(x+th)\)
Page: 460
Chapter: 8
Section: 4.3
Variables: \(m,k\in\mathbb N\)
\(G\subseteq\mathbb R^m\)
\(x\in G\), \(h\in\mathbb R^m\) such that \(x+h\in G\)
\(f: G\to \mathbb R\)
\(\phi(t) = f(x+th)\)
Index: 1
Kind: Ex
Smaller to greater: =
Assuming: \(f\in C^{(k)}(G, \mathbb R)\)
\(d\le k\)

Question: \(f(x^1+h^1,\dots,x^m+h^m\) equals
Answer: \(\sum_{k=0}^{n-1}\frac 1{k!}(\sum h^i\partial i)^kf(x) +\int_0^1\frac{(1-t)^{n-1}}{(n-1)!}(\sum_{i=1}^mh^i\partial_i)^nf(x+th)\mathtt{d}t\)
Page: 461
Chapter: 8
Section: 4.4
Variables: \(m,k\in\mathbb N\)
\(G\subseteq\mathbb R^m\)
\(x\in G\), \(h\in\mathbb R^m\) such that \(x+h\in G\)
\(f: G\to \mathbb R\)
\(\phi(t) = f(x+th)\)
Index: 4
Kind: Theo
Answer4: \(a\)
Assuming: \(f\in C^{(n)}(G, \mathbb R)\)
Name: Taylor's formula with integral form of the remainder

Question: \(f(x^1+h^1,\dots,x^m+h^m\) equals
Answer: \(\sum_{k=0}^{n-1}\frac 1{k!}(\sum h^i\partial i)^kf(x) +o\left(||h||^n\right)\)
Page: 461
Chapter: 8
Section: 4.4
Variables: \(m,k\in\mathbb N\)
\(G\subseteq\mathbb R^m\)
\(x\in G\), \(h\in\mathbb R^m\) such that \(x+h\in G\)
\(f: G\to \mathbb R\)
\(\phi(t) = f(x+th)\)
Index: 4
Kind: Theo
Assuming: \(f\in C^{(n)}(G, \mathbb R)\)
Name: Taylor's formula with the remainder term in Peano form

Name: A critical point of \(f\)
Definition: Some \(x\in G\) such that the rank of the Jacobi matrix at \(x\) is less than \(\min(m,n)\)
Page: 464
Chapter: 8
Section: 4.5
Variables: \(m\in\mathbb N\)
\(G\subseteq\mathbb R^m\)
\(f:G\to \mathbb R^n\)
Index: 3
Kind: Def
Example: all partial derivatives are 0
Applied to: \(n=1\)

Name: A stationary point of \(f\)
Definition: Some \(x\in G\) such that all partial derivatives are 0
Page: 464
Chapter: 8
Section: 4.5
Variables: \(m\in\mathbb N\)
\(G\subseteq\mathbb R^m\)
\(f:G\to \mathbb R\)
Index: 3
Kind: Def

Smaller0: the quadratic form \(\sum_{i,j=1}^m\partial_{i,j}f(x_0)h^ih^j\) is positive definite
Greater0: \(x_0\) is a strict local minimum
Page: 464
Chapter: 8
Section: 4.5
Variables: \(m\in\mathbb N\)
\(G\subseteq\mathbb R^m\)
\(x_0\in G\)
\(f:C^{(2)}(G;\mathbb R)\)
Index: 6
Kind: theo

Smaller0: the quadratic form \(\sum_{i,j=1}^m\partial_{i,j}f(x_0)h^ih^j\) is negative-definite
Greater0: \(x_0\) is a strict local maximum
Page: 464
Chapter: 8
Section: 4.5
Variables: \(m\in\mathbb N\)
\(G\subseteq\mathbb R^m\)
\(x_0\in G\)
\(f:C^{(2)}(G;\mathbb R)\)
Index: 6
Kind: theo

Smaller0: the quadratic form \(\sum_{i,j=1}^m\partial_{i,j}f(x_0)h^ih^j\) assumes both positive and negative values
Greater0: \(x_0\) is not an extremum
Page: 464
Chapter: 8
Section: 4.5
Variables: \(m\in\mathbb N\)
\(G\subseteq\mathbb R^m\)
\(x_0\in G\)
\(f:C^{(2)}(G;\mathbb R)\)
Index: 6
Kind: theo

Question: the minimal of \(\prod_{i=}^{n}\frac{m_{i}}{m_{i-1}+m_i}\)
Answer: \(m_i=m_0\left(\sqrt[n]{\frac{m_n}{m_0}}\right)^i\), i.e. a geometric progression of ratio \(\sqrt[n]{\frac{m_n}{m_0}}\)
Page: 469
Chapter: 8
Section: 4.5
Variables: \(m_0, m_n\mathbb R^{>0}\)
Index: 5
Kind: Ex
Name: The Huygens' problem
Intuition: Physically, if you want to transfer the maximal velocity from a ball of weight \(m_0\) to a ball of weight \(m_n\) with \(n-1\) intermediary balls, this is the optimal weight of those balls

Name: A coordinate line on \(S\)
Definition: A set of points \(\{f(x_0+t, y_0)\mid t\in\mathbb R\}\) or  \(\{f(x_0, y_0+t)\mid t\in\mathbb R\}\)
Page: 470
Chapter: 8
Section: 4.5
Variables: \(S\) a two-dimensional surface defined by \(f:\mathbb R^2\to\mathbb R\)

Smaller0: \(F\in C^{(p)}(U; \mathbb R)\)
Greater0: there exists intervals \(I_{x^1},\dots, X_{x^m}\) and \(I_y\) containing \(x^1,\dots,x^m\) and \(y\)
Page: 483, 486
Chapter: 8, 8
Section: 5, 5.3
Variables: \((x^1_0,\dots,x^m_0,y_0)\in\mathbb R^{m+1}\)
\(p\ge 1\)
\(U(x_0, y_0)\) a neighborhood of \((x_0, y_0)\)
\(F:U(x_0, y_0)\to\mathbb R\)
Index: 1
Kind: Prop
Assuming: n
Smaller1: \(F(x_0,y_0)=0\)
Name: Implicit function theorem
Greater1: there exists \(f\in C^{(p)}(I_x;I_y)\) such that
Smaller2: \(F'_y(x_0,y_0)\ne0\)
Greater2: \(F(x,y)=0\iff y=f(x)\) for any \(x\in I_x, y\in I_y\)
Greater3: \(\frac{\partial_i f}{\partial x^i}(x) = -\frac{F'_x(x, (f(x)))}{F'_y(x, f(x))}\)
Connect Greater2: and