hcju/proofwikips huggingface dataset:
Definition:Right Module
Let $\struct {R, +_R, \times_R}$ be a ring. Let $\struct {G, +_G}$ be an abelian group. A right module over $R$ is an $R$-algebraic structure $\struct {G, +_G, \circ}_R$ with one operation $\circ$, the '''(right) ring action''', which satisfies the right module axioms: {{:Definition:Right Module Axioms}}
Definition:O Notation/Little-O Notation/Sequence
Let $\left \langle {a_n} \right \rangle$ and $\left \langle {b_n} \right \rangle$ be sequences of real or complex numbers. === Definition 1 === {{:Definition:O Notation/Little-O Notation/Sequence/Definition 1}} === Definition 2 === {{:Definition:O Notation/Little-O Notation/Sequence/Definition 2}} This is denoted :$a_n = o \left({b_n}\right)$
Definition:O Notation/Little-O Notation/Sequence/Definition 1
Let $b_n\neq0$ for all $n$. '''$a_n$ is little-O of $b_n$'''  :$\displaystyle \lim_{n\to\infty}\frac{a_n}{b_n} = 0$
Definition:O Notation/Little-O Notation/Sequence/Definition 2
'''$a_n$ is little-O of $b_n$'''  :$\forall \epsilon \in \R: \epsilon > 0 : \exists n_0 \in \N : \left({n \ge n_0 \implies \left\vert{a_n}\right\vert \le \epsilon \cdot \left\vert{b_n}\right\vert}\right)$ That is: :For all $\epsilon>0$, $\left\vert{a_n}\right\vert \le \epsilon \cdot \left\vert{b_n}\right\vert$ for all sufficiently large $n$.
Definition:O Notation/Little-O Notation/Real Functions/Definition 1
Let $g(x)\neq0$ for $x$ sufficiently large. '''$f$ is little-o of $g$''' as $x \to \infty$ : :$\displaystyle \lim_{x \to \infty} \ \frac{f \left({x}\right)} {g \left({x}\right)} = 0$
Definition:O Notation/Little-O Notation/Real Functions/Definition 2
'''$f$ is little-o of $g$''' as $x \to \infty$ : :$\forall \epsilon \in \R : \epsilon > 0 : \exists x_0 \in \R : \forall x \geq x_0 : |f(x)| \leq \epsilon \cdot |g(x)|$
Definition:Ambivalent Group
Let $G$ be a group. Then $G$ is '''ambivalent'''  every element of $G$ is conjugate to its inverse: :$\forall g \in G : \exists h \in G : h g h^{-1} = g^{-1}$
Definition:Real Group Element
Let $G$ be a group. Let $g \in G$. Then $g$ is a '''real element (of $G$)'''  it is conjugate to its inverse: :$\exists h \in G : hgh^{-1} = g^{-1}$
Definition:O Notation/Big-O Notation/Uniform
Let $X$ be a set. Let $V$ be a normed vector space over $\R$ or $\C$ with norm $\norm {\, \cdot \,}$. Let $f, g : X \to V$ be mappings. Then '''$f$ is big $\OO$ of $g$ uniformly''' : :$\exists c > 0 : \forall x \in X : \norm {\map f x} \le c \cdot \norm {\map g x}$ This is denoted: :$f = \map \OO g$
Definition:Group Action/Right Group Action
A '''right group action''' is a mapping $\phi: X \times G \to X$ such that: :$\forall \tuple {x, g} \in X \times G : x * g := \map \phi {x, g} \in X$ in such a way that the right group action axioms are satisfied: {{:Definition:Right Group Action Axioms}}
Definition:Right Group Action Axioms
{{begin-axiom}} {{axiom | n = \text {RGA} 1         | q = \forall g, h \in G, x \in X         | m = \paren {x * g} * h = x * \paren {g \circ h} }} {{axiom | n = \text {RGA} 2         | q = \forall x \in X         | m =  x * e = x }} {{end-axiom}}
Definition:Conjugacy Action/Subgroups
Let $X$ be the set of all subgroups of $G$. The '''(left) conjugacy action on subgroups''' is the group action $* : G \times X \to X$: :$g * H = g \circ H \circ g^{-1}$ The '''right conjugacy action on subgroups''' is the group action $* : X \times G \to X$: :$H * g = g^{-1} \circ H \circ g$
Definition:Direct Sum of Module Homomorphisms
Let $R$ be a ring. Let $M,N,P,Q$ be $R$-modules. Let $M\oplus N$ and $P\oplus Q$ be their direct sum. Let $f : M \to P$ and $g : N \to Q$ be module homomorphisms. The '''direct sum''' of $f$ and $g$ is the module homomorphism $f\oplus g : M\oplus N \to P\oplus Q$ defined as: :$(f\oplus g) (m, n) = ( f(m) , g(n) )$
Definition:Transitive Group Action/n-transitive
Let $n\geq1$ be a natural number. The group action is '''$n$-transitive'''  for any two ordered $n$-tuples $(x_1, \ldots, x_n)$ and $(y_1, \ldots, y_n)$ of pairwise distinct elements of $S$, there exists $g\in G$ such that: :$\forall i\in \{1, \ldots, n\} : g * x_i = y_i$
Definition:Generated Submodule
=== Definition 1 === {{:Definition:Generated Submodule/Definition 1}} === Definition 2 === {{:Definition:Generated Submodule/Definition 2}}
Definition:Wilson Prime/Sequence
The sequence of Wilson primes begins: :$5, 13, 563$ The next term in the sequence, if there is one, is greater than $2 \times 10^{13}$.
Definition:Field Norm of Complex Number
Let $z = a + i b$ be a complex number, where $a, b \in \R$. Then the '''field norm of $z$''' is written $\map N z$ and is defined as: :$\map N z := \cmod \alpha^2 = a^2 + b^2$ where $\cmod \alpha$ denotes the complex modulus of $\alpha$.
Definition:Value of Continued Fraction/Finite
Let $n\geq0$ be a natural number. Let $(a_k)_{0 \leq k \leq n}$ be a finite continued fraction in $F$. Let $\overline F = F \cup \{\infty\}$ be extended by infinity. === Definition 1 === The '''value''' $[a_0, a_1, \ldots, a_n] \in F \cup \{\infty\}$ is the right iteration of the binary operation: :$[\cdot,\cdot] :  F \times \overline F \to \overline F$: :$[a, b] = a + \dfrac 1 b$. That is, it is recursively defined as: :$[a_0, \ldots, a_n] = \begin{cases}    a_0 & : n = 0 \\   a_0 + \dfrac 1 {\left[{a_1, \ldots, a_n}\right]} & : n > 0 \\ \end{cases}$ or as: :$[a_0, \ldots, a_n] = \begin{cases}    a_0 & : n = 0 \\   \left[a_0, \ldots, a_{n-2}, a_{n-1} + \dfrac 1 {a_n}\right] & : n > 0 \\ \end{cases}$ === Definition 2 === Let the matrix product: :$\begin{pmatrix}a_0 & 1 \\ 1 & 0\end{pmatrix}\cdots\begin{pmatrix}a_n & 1 \\ 1 & 0\end{pmatrix} = \begin{pmatrix}x_{11} & x_{12} \\ x_{21} & x_{22}\end{pmatrix}$ The '''value''' of the finite continued fraction is $\dfrac{x_{11}}{x_{21}}$
Definition:Value of Continued Fraction/Infinite
Let $\struct {F, \norm {\,\cdot\,}}$ be a valued field. Let $C = (a_n)_{n\geq 0}$ be a infinite continued fraction in $F$. Then $C$ '''converges''' to its '''value''' $x \in F$  the following hold: #For all natural numbers $n \in \N_{\geq 0}$, the $n$th denominator is nonzero #The sequence of convergents $\sequence{C_n}_{n\geq 0}$ converges to $x$.
Definition:Maximal Ideal of Ring/Left
A left ideal $J$ of $R$ is a '''maximal left ideal''' : :$(1): \quad J \subsetneq R$ :$(2): \quad$ There is no left ideal $K$ of $R$ such that $J \subsetneq K \subsetneq R$.
Definition:Maximal Ideal of Ring/Right
A right ideal $J$ of $R$ is a '''maximal right ideal''' : :$(1): \quad J \subsetneq R$ :$(2): \quad$ There is no right ideal $K$ of $R$ such that $J \subsetneq K \subsetneq R$.
Definition:Residue Field of Local Ring
Let $R$ be a commutative local ring. Let $m$ be its maximal ideal. The '''residue field''' is the quotient ring $R/m$.
Definition:Asymptotically Equal/Sequences/Definition 1
Let $b_n \ne 0$ for all $n$. $\sequence {a_n}$ is '''asymptotically equal''' to  $\sequence {b_n}$ : :$\displaystyle \lim_{n \to \infty} \dfrac {a_n} {b_n} \to 1$
Definition:Asymptotically Equal/Sequences/Definition 2
$\sequence {a_n}$ is '''asymptotically equal''' to  $\sequence {b_n}$ : :$a_n - b_n = \map o {b_n}$ where $o$ denotes little-o notation.
Definition:Asymptotically Equal/Sequences/Definition 3
$\sequence {a_n}$ is '''asymptotically equal''' to  $\sequence {b_n}$ : :$a_n - b_n = \map o {a_n}$ where $o$ denotes little-o notation.
Definition:Differential/Functional
Let $J \sqbrk y$ be a differentiable functional. Let $h$ be an increment of the independent variable $y$. Then the term linear  $h$ is called the '''differential''' of the functional $J$, and is denoted by $\delta J \sqbrk {y; h}$.
Definition:Compatible Atlases/Definition 1
$\mathscr F, \mathscr G$ are '''$C^k$-compatible'''  their union $\mathscr F \cup \mathscr G$ is an atlas of class $C^k$.
Definition:Compatible Atlases/Definition 2
$\mathscr F$ and $\mathscr G$ are '''$C^k$-compatible'''  every pair of charts $\struct {U, \phi} \in \mathscr F$ and $\struct {V, \psi} \in \mathscr G$ are $C^k$-compatible.
Definition:Atlas/Maximal Atlas
Let $A$ be a $d$-dimensional atlas of class $C^k$ of $M$. === Definition 1 === {{:Definition:Atlas/Maximal Atlas/Definition 1}} === Definition 2 === {{:Definition:Atlas/Maximal Atlas/Definition 2}} === Definition 3 === {{:Definition:Atlas/Maximal Atlas/Definition 3}}
Definition:Transition Mapping between Charts
Let $M$ be a topological space. Let $d$ be a natural number. Let $(U,\phi)$ and $(V,\psi)$ be $d$-dimensional charts of $M$. Let $U\cap V \neq\emptyset$. The '''transition map from $\phi$ to $\psi$''' is the mapping: :$\psi \circ \phi^{-1} : \phi(U\cap V) \to \psi(U\cap V)$
Definition:Compatible Charts
Let $M$ be a topological space. Let $d$ be a natural number. Let $\struct {U, \phi}$ and $\struct {V, \psi}$ be $d$-dimensional charts of $M$. Then $\struct {U, \phi}$ and $\struct {V, \psi}$ are '''$C^k$-compatible'''  their transition mapping: :$\psi \circ \phi^{-1}: \map \phi {U \cap V} \to \map \psi {U \cap V}$ is of class $C^k$. === Smoothly Compatible Charts === {{:Definition:Compatible Charts/Smooth}}
Definition:Compatible Charts/Smooth
$\struct {U, \phi}$ and $\struct {V, \psi}$ are '''smoothly compatible'''  their transition mapping: :$\psi \circ \phi^{-1} : \map \phi {U \cap V} \to \map \psi {U \cap V}$ is of class $C^\infty$.
Definition:Chart Compatible with Atlas
Let $M$ be a topological space. Let $A$ be a $d$-dimensional $C^k$-atlas on $M$. Let $\struct {U, \phi}$ be a $d$-dimensional chart of $M$. Then $\struct {U, \phi}$ is '''$C^k$-compatible with $A$'''  $\struct {U, \phi}$ is $C^k$-compatible with every chart of $A$.
Definition:Atlas/Maximal Atlas/Definition 1
$A$ is a '''maximal $C^k$-atlas''' of dimension $d$  $A$ is not strictly contained in another $C^k$-atlas.
Definition:Atlas/Maximal Atlas/Definition 2
$A$ is a '''maximal $C^k$-atlas'''  $A$ contains all charts of $M$ that are $C^k$-compatible with $A$.
Definition:Atlas/Maximal Atlas/Definition 3
$A$ is a '''maximal $C^k$-atlas'''  $A$ is a maximal element of some differentiable structure, partially ordered by inclusion. That is, a maximal element of some equivalence class of the set of atlases of class $\mathcal C^k$ on $M$ under the equivalence relation of compatibility.
Definition:Uniform Contraction Mapping
Let $M$ and $N$ be metric spaces. Let $f : M \times N \to M$ be a mapping. Then $f$ is a '''uniform contraction'''  there exists $K<1$ such that for all $x,y\in M$ and $t\in N$: :$d(f(x,t), f(y,t)) \leq K\cdot d(x,y)$.
Definition:Local Diffeomorphism
Let $n$ and $k$ be natural numbers. Let $U \subset \R^n$ be an open set. {{explain|Context of Open Set}} Let $f: U \to \R^n$ be a mapping. Then $f$ is a '''local $C^k$-diffeomorphism'''  every $a \in U$ has a open neighborhhood such that the restriction of $f$ to it is a $C^k$-diffeomorphism on its image. === Smooth Local Diffeomorphism ===
Definition:Open Set/Real Analysis/Real Numbers
Let $I \subseteq \R$ be a subset of the set of real numbers. Then $I$ is '''open (in $\R$)''' : : $\forall x_0 \in I: \exists \epsilon \in \R_{>0}: \openint {x_0 - \epsilon} {x_0 + \epsilon} \subseteq I$ where $\openint {x_0 - \epsilon} {x_0 + \epsilon}$ is an open interval. Note that $\epsilon$ may depend on $x_0$.
Definition:Open Set/Real Analysis/Real Euclidean Space
Let $n \ge 1$ be a natural number. Let $U \subseteq \R^n$ be a subset. Then $U$ is '''open (in $\R^n$)''' : :$\forall x \in U : \exists R > 0: \map B {x, R} \subset U$ where $\map B {x, R}$ denotes the open ball of radius $R$ centered at $x$.
Definition:Smith Number
A '''Smith number''' is a composite number for which the sum of its digits is equal to the sum of the digits in its prime decomposition.
Definition:Smith Brothers
'''Smith brothers''' are two consecutive integers which are both '''Smith numbers'''. === Sequence of Smith Brothers === {{:Definition:Smith Brothers/Sequence}}
Definition:Differentiable Mapping/Real Function/Point/Definition 1
$f$ is '''differentiable at the point $\xi$'''  the limit: :$\displaystyle \lim_{x \mathop \to \xi} \frac {\map f x - \map f \xi} {x - \xi}$ exists.
Definition:Differentiable Mapping/Real Function/Point/Definition 2
$f$ is '''differentiable at the point $\xi$'''  the limit: :$\displaystyle \lim_{h \mathop \to 0} \frac {\map f {\xi + h} - \map f \xi} h$ exists.
Definition:Open Ball/Real Analysis
Let $n \ge 1$ be a natural number. Let $\R^n$ denote a real Euclidean space Let $\left\Vert{\cdot}\right\Vert$ denote the Euclidean norm. Let $a \in \R^n$. Let $R > 0$ be a strictly positive real number. The '''open ball of center $a$ and radius $R$''' is the subset: :$B \left({a, R}\right) = \left\{ {x \in \R^n : \left\Vert{x - a}\right\Vert < R}\right\}$
Definition:Closed Rectangle
Let $n\geq1$ be a natural number. Let $a_1, \ldots, a_n, b_1, \ldots, b_n$ be real numbers. The Cartesian product of closed intervals: :$\displaystyle \prod_{i \mathop = 1}^n \left[{a_i \,.\,.\, b_i}\right] = \left[{a_1 \,.\,.\, b_1}\right] \times \cdots \times \left[{a_n \,.\,.\, b_n}\right] \subseteq \R^n$ is called a '''closed rectangle in $\R^n$''' or '''closed $n$-rectangle'''. === Degenerate Case === In case $a_i > b_i$ for some $i$, the rectangle is taken to be the empty set $\varnothing$. This is in accordance with the result Cartesian Product is Empty iff Factor is Empty for general Cartesian products.
Definition:Closed Ball/Real Analysis
Let $n \ge 1$ be a natural number. Let $\R^n$ denote real Euclidean space Let $\left\Vert{\, \cdot \,}\right\Vert$ denote the Euclidean norm. Let $a \in \R^n$. Let $R > 0$ be a strictly positive real number. The '''closed ball of center $a$ and radius $R$''' is the subset: :$B \left({a, R}\right) = \left\{ {x \in \R^n : \left\Vert{x - a}\right\Vert \le R}\right\}$
Definition:Closed Set/Real Analysis/Real Numbers
Let $S \subseteq \R$ be a subset of the set of real numbers. Then $S$ is '''closed (in $\R$)'''  its complement $\R \setminus S$ is an open set.
Definition:Closed Set/Real Analysis/Real Euclidean Space
Let $n\geq1$ be a natural number. Let $S \subseteq \R^n$ be a subset. Then $S$ is '''closed (in $\R^n$)'''  its complement $\R^n \setminus S$ is an open set.
Definition:Topology Generated by Synthetic Basis/Definition 1
The '''topology on $S$ generated by $\BB$''' is defined as: :$\tau = \set{\bigcup \AA: \AA \subseteq \BB}$
Definition:Topology Generated by Synthetic Basis/Definition 2
The '''topology on $S$ generated by $\BB$''' is defined as: :$\tau = \set {U \subseteq S: U = \bigcup \set {B \in \BB: B \subseteq U}}$
Definition:Basis (Topology)/Synthetic Basis/Definition 1
A '''synthetic basis on $S$''' is a subset $\mathcal B \subseteq \mathcal P \left({S}\right)$ of the power set of $S$ such that: {{begin-axiom}} {{axiom | n = B1         | t = $\mathcal B$ is a cover for $S$ }} {{axiom | n = B2         | q = \forall U, V \in \mathcal B         | t = $\exists \mathcal A \subseteq \mathcal B: U \cap V = \bigcup \mathcal A$ }} {{end-axiom}} That is, the intersection of any pair of elements of $\mathcal B$ is a union of sets of $\mathcal B$.
Definition:Basis (Topology)/Synthetic Basis/Definition 2
A '''synthetic basis on $S$''' is a subset $\mathcal B \subseteq \mathcal P \left({S}\right)$ of the power set of $S$ such that: : $\mathcal B$ is a cover for $S$ : $\forall U, V \in \mathcal B: \forall x \in U \cap V: \exists W \in \mathcal B: x \in W \subseteq U \cap V$
Definition:Topology Induced by Atlas
Let $X$ be a set. Let $A$ be an atlas on $X$. The '''topology induced by the atlas''' is the topology generated by the synthetic sub-basis: :$\displaystyle\bigcup_{(U,\phi) \in A }\left\{\phi^{-1}(V) : V \text{ is open in } U \right\} \subset X$
Definition:Generated Submodule/Definition 1
The '''submodule generated by $S$''' is the intersection of all submodules of $M$ containing $S$.
Definition:Generated Submodule/Definition 2
The '''submodule generated by $S$''' is the set of all linear combinations of elements of $S$.
Definition:Generator of Module/Definition 1
$S$ is a '''generator of $M$'''  every element of $M$ is a linear combination of elements of $S$.
Definition:Generator of Module/Definition 2
$S$ is a '''generator of $M$'''  $M$ has no proper submodule containing $S$.
Definition:Generator of Module/Definition 3
$S$ is a '''generator of $M$'''  $M$ is the submodule generated by $S$.
Definition:Generator of Vector Space
Let $K$ be a division ring. Let $\mathbf V$ be a vector space over $K$. Let $S \subseteq \mathbf V$ be a subset of $\mathbf V$. $S$ is a '''generator of $\mathbf V$'''  every element of $\mathbf V$ is a linear combination of elements of $S$.
Definition:Real Submanifold
Let $n,k\geq1$ be natural numbers. Let $M\subset\R^n$ be a subset. === Definition Using Local Diffeomorphisms === $M$ is a '''real $C^k$-submanifold of dimension $d$''' of $\R^n$  for all $p\in M$ there exists a open neighborhood $U$ of $p$ in $\R^n$ and a differentiable function $\phi : U \to \R^n$ that is a $C^k$-diffeomorphism on its image, such that: :$\phi(M \cap U) = \phi(U) \cap (\R^d\times\{0\})$ === Definition Using Local Submersions === $M$ is a '''real $C^\infty$-submanifold of dimension $d$''' of $\R^n$  for all $p\in M$ there exists a open neighborhood $U$ of $p$ in $\R^n$ and a $C^\infty$-submersion $\phi : U \to \R^{n-d}$ such that: :$M \cap U = \phi^{-1}(0)$ === Definition Using Local Embeddings === $M$ is a '''real $C^\infty$-submanifold of dimension $d$''' of $\R^n$  for all $p\in M$ there exists an open neighborhood $U$ in $\R^d$ and a $C^\infty$-embedding $\phi : U \to \R^{n}$ such that: :$p \in \phi(U) \subset M$
Definition:Embedding (Differential Geometry)
Let $m,n\geq1$ be natural numbers. Let $U\subset\R^n$ be open. Let $f : U \to \R^m$ be a mapping. Then $f$ is a $C^k$-'''embedding'''  $f$ is: * injective * a $C^k$-immersion * a homeomorphism on its image === Rank === The '''rank''' of an embedding is the rank of its differential at any point. === Smooth Embedding ===
Definition:Open Neighborhood/Real Analysis
=== Real Numbers === {{:Definition:Open Neighborhood/Real Analysis/Real Numbers}} === Real Euclidean Space === {{:Definition:Open Neighborhood/Real Analysis/Real Euclidean Space}}
Definition:Open Neighborhood/Real Analysis/Real Numbers
Let $x\in\R$ be a real number. Let $I \subseteq \R$ be a subset. Then $I$ is an '''open neighborhood of $x$'''  $I$ is open and $I$ is a neighborhood of $x$.
Definition:Continuously Differentiable/Real Function
=== On an Open Interval === {{:Definition:Continuously Differentiable/Real Function/Open Interval}} === On an Open Set === {{:Definition:Continuously Differentiable/Real Function/Open Set}}
Definition:Continuously Differentiable/Real Function/Open Interval
Let $I\subset\R$ be an open interval. Then $f$ is '''continuously differentiable on $I$'''  $f$ is differentiable on $I$ and its derivative is continuous on $I$.
Definition:Continuously Differentiable/Real Function/Open Set
Let $I \subset \R$ be an open set. Then $f$ is '''continuously differentiable on $I$'''  $f$ is differentiable on $I$ and its derivative is continuous on $I$.
Definition:Continuously Differentiable/Real-Valued Function
=== In an Open Set === {{:Definition:Continuously Differentiable/Real-Valued Function/Open Set}}
Definition:Continuously Differentiable/Real-Valued Function/Open Set
Let $U$ be an open subset of $\R^n$.  Let $f: U \to \R$ be a real-valued function. Then $f$ is '''continuously differentiable in the open set $U$''' : :$(1): \quad f$ is differentiable in $U$. :$(2): \quad$ the partial derivatives of $f$ are continuous in $U$.
Definition:Continuously Differentiable/Vector-Valued Function
=== On an Open Set === {{:Definition:Continuously Differentiable/Vector-Valued Function/Open Set}}
Definition:Continuously Differentiable/Vector-Valued Function/Open Set
Let $U \subset \R^n$ be an open set. Let $f: U \to \R^m$ be a vector-valued function. Then $f$ is '''continuously differentiable in $U$'''  $f$ is differentiable in $U$ and its partial derivatives are continuous in $U$.
Definition:Holomorphic Function/Complex Plane
Let $U\subset\C$ be an open set. Let $f : U \to \C$ be a complex function. Then $f$ is '''holomorphic in $U$'''  $f$ is differentiable at each point of $U$.
Definition:Derivative/Complex Function/Point
Let $D\subseteq \C$ be an open set. Let $f : D \to \C$ be a complex function. Let $z_0 \in D$ be a point in $D$. Let $f$ be complex-differentiable at the point $z_0$. That is, suppose the limit $\displaystyle \lim_{h \to 0} \ \frac {f \left({z_0 + h}\right) - f \left({z_0}\right)} h$ exists. Then this limit is called the '''derivative of $f$ at the point $z_0$'''.
Definition:Derivative/Complex Function/Open Set
Let $D\subseteq \C$ be an open set. Let $f : D \to \C$ be a complex function. Let $f$ be complex-differentiable in $D$. Then the '''derivative of $f$''' is the complex function  $f': D \to \C$ whose value at each point $z \in D$ is the derivative $f' \left({z}\right)$: :$\displaystyle \forall z \in D : f' \left({z}\right) := \lim_{h \mathop \to 0} \frac {f \left({z + h}\right) - f \left({z}\right)} h$
Definition:Differentiable Mapping/Vector-Valued Function/Point/Definition 1
$f$ is '''differentiable''' at $x \in \R^n$  there exists a linear transformation $T: \R^n \to \R^m$ and a mapping $r : U \to \R^m$ such that: :$(1): \quad \map f {x + h} = \map f x + \map T h + \map r h \cdot \norm h$ :$(2): \quad \displaystyle \lim_{h \mathop \to 0} \map r h = 0$
Definition:Differentiable Mapping/Vector-Valued Function/Point/Definition 2
$f$ is '''differentiable''' at $x \in \R^n$  for each real-valued function $f_j: j = 1, 2, \ldots, m$: :$f_j: \mathbb X \to \R$ is differentiable at $x$.
Definition:Differential/Real Function/Point
Let $U \subset \R$ be an open set. Let $f: U \to \R$ be a real function. Let $f$ be differentiable at a point $x \in U$. The '''differential of $f$ at $x$''' is the linear transformation $\rd f \left({x}\right) : \R \to \R$ defined as: :$\rd f \left({x}\right) \left({h}\right) = f' \left({x}\right) \cdot h$ where $f' \left({x}\right)$ is the derivative of $f$ at $x$.
Definition:Differential/Real Function/Open Set
Let $U \subset \R$ be an open set. Let $f : U \to \R$ be a real function. Let $f$ be differentiable in $U$. The '''differential''' $\rd f$ is the mapping $\rd f : U \to \operatorname{Hom} \left({\R, \R}\right)$ defined as: :$\left({\mathrm d f}\right) \left({x}\right) = \rd f \left({x}\right)$ where: :$\rd f \left({x}\right)$ is the differential of $f$ at $x$ :$\operatorname{Hom} \left({\R, \R}\right)$ is the set of all linear transformations from $\R$ to $\R$.
Definition:Zuckerman Number
A '''Zuckerman number''' is a positive integer which is divisible by the product of its digits.
Definition:Partial Derivative/Real Analysis
{{:Definition:Partial Derivative/Real Analysis/Point}}
Definition:Partial Derivative/Real Analysis/Point
Let $U \subset \R^n$ be an open set. Let $f: U \to \R$ be a real-valued function. Let $a = \tuple {a_1, a_2, \ldots, a_n}^\intercal \in U$. Let $f$ be differentiable at $a$. Let $i \in \set {1, 2, \ldots, n}$. ==== Definition 1 ==== {{:Definition:Partial Derivative/Real Analysis/Point/Definition 1}} ==== Definition 2 ==== {{:Definition:Partial Derivative/Real Analysis/Point/Definition 2}}
Definition:Partial Derivative/Real Analysis/Open Set
Let $U\subset\R^n$ be an open set. Let $f : U \to \R$ be a real-valued function. Let $f$ be differentiable in $U$. The '''$i$th partial derivative (function) of $f$ with respect to $x_i$''' is the real-valued function which sends each $x\in U$ to the $i$th partial derivative at $x$.
Definition:Partial Derivative/Real Analysis/Point/Definition 1
The '''partial derivative of $f$ with respect to $x_i$ at $a$''' is denoted and defined as: :$\map {\dfrac {\partial f} {\partial x_i} } a := \map {g_i'} {a_i}$ where: :$g_i$ is the real function defined as $\map g {x_i} = \map f {a_1, \ldots, x_i, \dots, a_n}$ :$\map {g_i'} {a_i}$ is the derivative of $g$ at $a_i$.
Definition:Partial Derivative/Real Analysis/Point/Definition 2
The '''$i$th partial derivative of $f$ at $a$''' is the limit: :$\map {\dfrac {\partial f} {\partial x_i} } a = \displaystyle \lim_{x_i \mathop \to a_i} \frac {\map f {a_1, a_2, \ldots, x_i, \ldots, a_n} - \map f a} {x_i - a}$