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Coordinatewise Characterization of Continuity for Euclidean Maps
theoremthm:continuous-coordinatewise-euclidean-2026aMultivariable Calculus
Let $n,m\in\mathbb{N}$ . Let $E\subseteq \mathbb{R}^n$ , let $f=(f_1,\dots,f_m):E\to\mathbb{R}^m$ , and let $a\in E$ . Then the following are equivalent. The map $f$ is \reftext{def:continuous-map-at-point-euclidean-2026a}{continuous} at $a$ . For every index $j\in\{1,\dots,m\}$ , the coordinate function $f_j:E\to\mathbb{R}$ is continuous at $a$ in the sense of \reftext{def:continuous-at-point-c54-2026b}{continuity for real-valued functions}.
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T21:14:19.736365Z
Finite Product Notation
definitiondef:finite-product-notation-2026aCombinatorics
Let $n\in\mathbb{N}$ , and let $a_1,\dots,a_n\in\mathbb{R}$ . The finite product $\prod_{i=1}^n a_i$ is defined recursively as follows. $\prod_{i=1}^1 a_i=a_1.$ For every natural number $n\ge 2$ , one sets $\prod_{i=1}^n a_i=\left(\prod_{i=1}^{n-1} a_i\right)a_n.$
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:51:48.637636Z
Bijection of Sets
definitiondef:bijection-sets-2026aCombinatorics
Let $X$ and $Y$ be sets. A bijection from $X$ to $Y$ is a function $f:X\to Y$ with the following property: for every element $y\in Y$ there exists exactly one element $x\in X$ such that $f(x)=y.$
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:51:48.589180Z
Sign of a Permutation
definitiondef:sign-permutation-2026aCombinatorics
Let $r\in\mathbb{N}$ , and let $\sigma\in S_r$ , where $S_r$ is the set from \reftext{def:permutation-initial-segment-2026a}{the permutation definition}. An inversion of $\sigma$ is a pair $(i,j)$ such that $1\le i<j\le r$ and $\sigma(i)>\sigma(j)$ . Let $N(\sigma)$ denote the number of inversions of $\sigma$ . The sign of $\sigma$ is the number $\operatorname{sgn}(\sigma)$ defined by $\operatorname{sgn}(\sigma)=1$ if $N(\sigma)$ is \reftext{def:even-odd-natural-numbers-2026a}{even}, and by $\operatorname{sgn}(\sigma)=-1$ if $N(\sigma)$ is \reftext{def:even-odd-natural-numbers-2026a}{odd}.
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:44:32.767762Z
Permutation of the Set $\{1,\dots,r\}$
definitiondef:permutation-initial-segment-2026aCombinatorics
Let $r\in\mathbb{N}$ . A permutation of the set $\{1,\dots,r\}$ is a \reftext{def:bijection-sets-2026a}{bijection} $\sigma:\{1,\dots,r\}\to\{1,\dots,r\}.$ The set of all permutations of $\{1,\dots,r\}$ is denoted by $S_r$ .
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:44:32.722421Z
Even and Odd Natural Numbers
definitiondef:even-odd-natural-numbers-2026aCombinatorics
Let $n\in\mathbb{N}$ . We say that $n$ is even if there exists a natural number $q\in\mathbb{N}$ such that $n=2q.$ We say that $n$ is odd if there exists a natural number $q\in\mathbb{N}$ such that $n=2q-1.$
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:44:32.676291Z
Factorial of a Natural Number
definitiondef:factorial-natural-number-2026aCombinatorics
Let $n\in\mathbb{N}$ . The factorial of $n$ is the natural number $n!=\prod_{i=1}^n i,$ where the product is the finite product from \reftext{def:finite-product-notation-2026a}{the finite-product definition}.
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:44:32.622571Z
Matrix-Vector Product
definitiondef:matrix-vector-product-2026aLinear Algebra
Let $m,n\in\mathbb{N}$ . Let $A=(A_{\alpha i})$ be an $m\times n$ matrix with real entries, and let $v=(v_1,\dots,v_n)\in\mathbb{R}^n$ . The product vector $Av\in\mathbb{R}^m$ is defined by $(Av)_\alpha=\sum_{i=1}^n A_{\alpha i}v_i$ for every $\alpha\in\{1,\dots,m\}$ .
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:40:23.168187Z
Cartesian Product of Sets
definitiondef:cartesian-product-sets-2026aCombinatorics
Let $X$ and $Y$ be sets. Their Cartesian product is the set $X\times Y=\{(x,y): x\in X \text{ and } y\in Y\}.$
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:40:22.898067Z
Product of Real Matrices
definitiondef:product-real-matrices-2026aLinear Algebra
Let $m,n,p\in \mathbb{N}$ . Let $A=(A_{\alpha i})$ be an $m\times n$ matrix with real entries, and let $B=(B_{i\beta})$ be an $n\times p$ matrix with real entries. The product matrix $AB$ is the $m\times p$ matrix whose $(\alpha,\beta)$ entry is defined by $(AB)_{\alpha\beta}=\sum_{i=1}^n A_{\alpha i}B_{i\beta}$ for every $\alpha\in\{1,\dots,m\}$ and every $\beta\in\{1,\dots,p\}$ .
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:30:26.036407Z
Euclidean Space $\mathbb{R}^n$
definitiondef:euclidean-space-rn-2026aMultivariable Calculus
Let $n\in \mathbb{N}$ . The set of all ordered $n$ -tuples $(x_1,\dots,x_n)$ with each $x_i\in \mathbb{R}$ is denoted by $\mathbb{R}^n$ and is called $n$ -dimensional Euclidean space.
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:27:09.013170Z
Natural Numbers
definitiondef:natural-numbers-2026aCombinatorics
We write $\mathbb{N}=\{1,2,3,\dots\}.$ The elements of $\mathbb{N}$ are called natural numbers. We regard addition and multiplication on $\mathbb{N}$ as binary operations $+ : \mathbb{N}\times\mathbb{N}\to\mathbb{N} \quad\text{and}\quad \cdot : \mathbb{N}\times\mathbb{N}\to\mathbb{N},$ written in infix form as $(m,n)\mapsto m+n$ and $(m,n)\mapsto mn$ . We also regard the successor on $\mathbb{N}$ as a function $S:\mathbb{N}\to\mathbb{N}$ written in infix form as $n\mapsto S(n)$ . These data are related by the following recursive identities for all $m,n\in\mathbb{N}$ . $S(n)=n+1$ . $m+S(n)=S(m+n)$ . $m\cdot 1=m$ . $m\cdot S(n)=m\cdot n+m$ . We also require that $1$ is not a successor and that the successor map is injective; that is, $S(n)\ne 1\quad\text{for every } n\in\mathbb{N},$ and $S(m)=S(n)\implies m=n$ for all $m,n\in\mathbb{N}$ .
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:27:08.937207Z
Chain Rule for $C^1$ Maps Between Euclidean Spaces
theoremthm:chain-rule-c1-euclidean-2026aMultivariable Calculus
Let $n,m,p\in \mathbb{N}$ . Let $U\subseteq \mathbb{R}^n$ , $V\subseteq \mathbb{R}^m$ , and $W\subseteq \mathbb{R}^p$ be \reftext{def:open-subset-euclidean-space-2026a}{open} subsets. Let $f:U\to V$ and $g:V\to W$ be \reftext{def:c1-map-euclidean-open-set-2026a}{ $C^1$ maps}. Then the composition $g\circ f:U\to W$ is again of class $C^1$ . Moreover, for every $a\in U$ , the Jacobian matrices from \reftext{def:differentiable-map-at-point-euclidean-2026a}{the differentiability definition} satisfy $J_{g\circ f}(a)=J_g(f(a))\,J_f(a),$ where the product on the right-hand side is the matrix product from \reftext{def:product-real-matrices-2026a}{the definition of matrix multiplication}.
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:02:45.863686Z
Pullback of a Differential Form by a $C^1$ Map
definitiondef:pullback-differential-form-c1-euclidean-2026aGeometryMultivariable Calculus
Let $n,m,k\in\mathbb{N}$ . Let $U\subseteq \mathbb{R}^n$ and $V\subseteq \mathbb{R}^m$ be \reftext{def:open-subset-euclidean-space-2026a}{open} subsets, let $F:U\to V$ be a \reftext{def:c1-map-euclidean-open-set-2026a}{ $C^1$ map}, and let $\omega$ be a \reftext{def:differential-k-form-euclidean-open-set-2026a}{differential $k$ -form} on $V$ . The pullback of $\omega$ by $F$ is the differential $k$ -form $F^{*}\omega$ on $U$ defined by $(F^{*}\omega)_x(v_1,\dots,v_k)=\omega_{F(x)}\bigl(J_F(x)v_1,\dots,J_F(x)v_k\bigr)$ for every $x\in U$ and every vectors $v_1,\dots,v_k\in\mathbb{R}^n$ , where $J_F(x)v_r$ denotes the matrix-vector product from \reftext{def:matrix-vector-product-2026a}{the definition of matrix-vector multiplication}, applied to the Jacobian matrix appearing in \reftext{def:differentiable-map-at-point-euclidean-2026a}{the differentiability definition}.
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:02:45.816753Z
Wedge Product of Differential Forms on Euclidean Space
definitiondef:wedge-product-differential-forms-euclidean-2026bGeometryMultivariable Calculus
Let $n\in\mathbb{N}$ and let $k,\ell\in\mathbb{N}\cup\{0\}$ . Let $U\subseteq \mathbb{R}^n$ be \reftext{def:open-subset-euclidean-space-2026a}{open}. Let $\alpha$ be a \reftext{def:differential-k-form-euclidean-open-set-2026a}{differential $k$ -form} on $U$ , and let $\beta$ be a differential $\ell$ -form on $U$ . The wedge product $\alpha\wedge\beta$ is the differential $(k+\ell)$ -form on $U$ defined as follows: for each $x\in U$ and each collection of vectors $v_1,\dots,v_{k+\ell}\in\mathbb{R}^n$ , $(\alpha\wedge\beta)_x(v_1,\dots,v_{k+\ell}) =\frac{1}{k!\,\ell!}\sum_{\sigma\in S_{k+\ell}} \operatorname{sgn}(\sigma)\, \alpha_x(v_{\sigma(1)},\dots,v_{\sigma(k)})\, \beta_x(v_{\sigma(k+1)},\dots,v_{\sigma(k+\ell)}),$ where $S_{k+\ell}$ is the set from \reftext{def:permutation-initial-segment-2026a}{the definition of permutations}, $\operatorname{sgn}(\sigma)$ is the sign from \reftext{def:sign-permutation-2026a}{the sign definition}, and one uses the convention $0!=1$ . If $r\in\mathbb{N}$ and $\omega_1,\dots,\omega_r$ are differential forms on $U$ such that each successive wedge product is defined, then $\omega_1\wedge\cdots\wedge\omega_r$ means the left-associated iterated wedge product $(((\omega_1\wedge\omega_2)\wedge\omega_3)\wedge\cdots)\wedge\omega_r.$
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:02:45.770894Z
Differential k-Form on an Open Subset of Euclidean Space
definitiondef:differential-k-form-euclidean-open-set-2026aGeometryMultivariable Calculus
Let $n\in\mathbb{N}$ , let $U\subseteq \mathbb{R}^n$ be \reftext{def:open-subset-euclidean-space-2026a}{open}, and let $k\in\mathbb{N}\cup\{0\}$ . A differential $k$ -form on $U$ is an assignment $\omega$ which to each point $x\in U$ assigns an \reftext{def:alternating-k-linear-form-euclidean-2026a}{alternating $k$ -linear form} $\omega_x$ on $\mathbb{R}^n$ . For vectors $v_1,\dots,v_k\in\mathbb{R}^n$ , the value of $\omega$ at $x$ on $(v_1,\dots,v_k)$ is denoted by $\omega_x(v_1,\dots,v_k).$ When $k=0$ , this means exactly that a differential $0$ -form on $U$ is a real-valued function on $U$ .
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:02:45.729237Z
Alternating k-Linear Form on Euclidean Space
definitiondef:alternating-k-linear-form-euclidean-2026aGeometryMultivariable Calculus
Let $n,k\in\mathbb{N}$ . A function $\omega:(\mathbb{R}^n)^k\to\mathbb{R}$ is called a $k$ -linear form on $\mathbb{R}^n$ if for each index $r\in\{1,\dots,k\}$ , for every choice of vectors $v_1,\dots,v_{r-1},u,w,v_{r+1},\dots,v_k\in\mathbb{R}^n$ , and for every scalars $\alpha,\beta\in\mathbb{R}$ , one has $\omega(v_1,\dots,v_{r-1},\alpha u+\beta w,v_{r+1},\dots,v_k) =\alpha\,\omega(v_1,\dots,v_{r-1},u,v_{r+1},\dots,v_k) +\beta\,\omega(v_1,\dots,v_{r-1},w,v_{r+1},\dots,v_k).$ A $k$ -linear form $\omega$ is called alternating if $\omega(v_1,\dots,v_k)=0$ whenever $v_p=v_q$ for some distinct indices $p,q\in\{1,\dots,k\}$ . An alternating $k$ -linear form on $\mathbb{R}^n$ is also called an alternating covariant $k$ -tensor on $\mathbb{R}^n$ .
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:02:45.693210Z
$C^1$ Map on an Open Subset of Euclidean Space
definitiondef:c1-map-euclidean-open-set-2026aMultivariable Calculus
Let $n,m\in\mathbb{N}$ . Let $U\subseteq \mathbb{R}^n$ be \reftext{def:open-subset-euclidean-space-2026a}{open}, and let $f=(f_1,\dots,f_m):U\to\mathbb{R}^m$ . We say that $f$ is of class $C^1$ on $U$ if each coordinate function $f_j:U\to\mathbb{R}$ is \reftext{def:continuous-map-at-point-euclidean-2026a}{continuous at every point of $U$ }, and if for every $j\in\{1,\dots,m\}$ and every $i\in\{1,\dots,n\}$ the partial derivative \ref{def:partial-derivative-coordinate-map-2026a} $\frac{\partial f_j}{\partial x_i}(x)$ exists for every $x\in U$ , with the function $x\mapsto \frac{\partial f_j}{\partial x_i}(x)$ from $U$ to $\mathbb{R}$ also \reftext{def:continuous-map-at-point-euclidean-2026a}{continuous at every point of $U$ }.
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:02:45.653161Z
Differentiability at a Point and Jacobian Matrix for Maps Between Euclidean Spaces
definitiondef:differentiable-map-at-point-euclidean-2026aMultivariable Calculus
Let $n,m\in\mathbb{N}$ . Let $U\subseteq \mathbb{R}^n$ be \reftext{def:open-subset-euclidean-space-2026a}{open}, let $f=(f_1,\dots,f_m):U\to\mathbb{R}^m$ , and let $a=(a_1,\dots,a_n)\in U$ . Suppose that for every $j\in\{1,\dots,m\}$ and every $i\in\{1,\dots,n\}$ the partial derivative \ref{def:partial-derivative-coordinate-map-2026a} $\frac{\partial f_j}{\partial x_i}(a)$ exists. The matrix $J_f(a)=\left(\frac{\partial f_j}{\partial x_i}(a)\right)_{1\le j\le m,\ 1\le i\le n}$ is called the Jacobian matrix of $f$ at $a$ . We say that $f$ is differentiable at $a$ if for every $\varepsilon>0$ there exists $\delta>0$ with the following property: whenever $h=(h_1,\dots,h_n)\in\mathbb{R}^n$ satisfies $0<\sum_{i=1}^n h_i^2<\delta^2$ and $a+h\in U$ , one has $\sum_{j=1}^m \left(f_j(a+h)-f_j(a)-\sum_{i=1}^n \frac{\partial f_j}{\partial x_i}(a) h_i\right)^2 \le \varepsilon^2 \sum_{i=1}^n h_i^2.$
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:02:45.560180Z
Partial Derivative of a Coordinate Function
definitiondef:partial-derivative-coordinate-map-2026aMultivariable Calculus
Let $n,m\in\mathbb{N}$ . Let $U\subseteq \mathbb{R}^n$ be \reftext{def:open-subset-euclidean-space-2026a}{open}, let $f=(f_1,\dots,f_m):U\to\mathbb{R}^m$ , let $a=(a_1,\dots,a_n)\in U$ , and fix indices $i\in\{1,\dots,n\}$ and $j\in\{1,\dots,m\}$ . We say that the partial derivative of the $j$ th coordinate function of $f$ with respect to the $i$ th variable exists at $a$ if there exists a real number $L$ such that for every $\varepsilon>0$ there exists $\delta>0$ with the following property: whenever $h\in\mathbb{R}$ satisfies $0<|h|<\delta$ , one has $\left|\frac{f_j(a_1,\dots,a_{i-1},a_i+h,a_{i+1},\dots,a_n)-f_j(a_1,\dots,a_n)}{h}-L\right|<\varepsilon.$ In that case $L$ is called the partial derivative of $f_j$ with respect to $x_i$ at $a$ and is denoted by $\frac{\partial f_j}{\partial x_i}(a).$
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:02:45.507516Z

Showing 161-180 of 321

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Coordinatewise Characterization of Continuity for Euclidean Maps

Finite Product Notation

Bijection of Sets

Sign of a Permutation

Permutation of the Set {1,…,r}\{1,\dots,r\}{1,…,r}

Even and Odd Natural Numbers

Factorial of a Natural Number

Matrix-Vector Product

Cartesian Product of Sets

Product of Real Matrices

Euclidean Space Rn\mathbb{R}^nRn

Natural Numbers

Chain Rule for C1C^1C1 Maps Between Euclidean Spaces

Pullback of a Differential Form by a C1C^1C1 Map

Wedge Product of Differential Forms on Euclidean Space

Differential k-Form on an Open Subset of Euclidean Space

Alternating k-Linear Form on Euclidean Space

C1C^1C1 Map on an Open Subset of Euclidean Space

Differentiability at a Point and Jacobian Matrix for Maps Between Euclidean Spaces

Partial Derivative of a Coordinate Function

Permutation of the Set $\{1,\dots,r\}$

Euclidean Space $\mathbb{R}^n$

Chain Rule for $C^1$ Maps Between Euclidean Spaces

Pullback of a Differential Form by a $C^1$ Map

$C^1$ Map on an Open Subset of Euclidean Space