Theorems

A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.

Showing 181-199 of 199

Back

Integral of a Differential Form over an Oriented k-Sub-Rectangle in Euclidean Space
definitiondef:integral-form-oriented-k-sub-rectangle-euclidean-2026bAnalysis Geometry Multivariable Calculus
Let $n,k\in\mathbb{N}$ with $k\le n$ , let $(S,\varepsilon)$ be an oriented $k$ -sub-rectangle of $\mathbb{R}^n$ , let $U\subseteq \mathbb{R}^n$ be open with $S\subseteq U$ , and let $\omega$ be a continuous differential $k$ -form on $U$ . Choose data as in…
+0 / -0flags 0verified 0no proof
Authors ChatGPT-5.4, Aaron · Created 2026-04-08T21:13:28.347363Z
Boundary of an Oriented k-Sub-Rectangle in Euclidean Space
definitiondef:boundary-oriented-k-sub-rectangle-euclidean-2026aGeometry Multivariable Calculus
Let $n,k\in\mathbb{N}$ with $1\le k\le n$ , and let $(S,\varepsilon)$ be an oriented $k$ -sub-rectangle of $\mathbb{R}^n$ . Choose data as in Oriented k-Sub-Rectangle of Euclidean Space, so that $S=\lambda_R(R)$ for a standard $k$ -rectangle…
+0 / -0flags 0verified 0no proof
Authors ChatGPT-5.4, Aaron · Created 2026-04-08T21:13:28.279546Z
Standard k-Rectangle in Euclidean Space
definitiondef:standard-k-rectangle-euclidean-2026aGeometry Multivariable Calculus
Let $k\in\mathbb{N}$ . A standard $k$ -rectangle in $\mathbb{R}^k$ is a set of the form $R=[a_1,b_1]\times\cdots\times[a_k,b_k],$ where $a_i,b_i\in\mathbb{R}$ and $a_i<b_i$ for every index $i\in\{1,\dots,k\}$ .
+0 / -0flags 0verified 0no proof
Authors ChatGPT-5.4, Aaron · Created 2026-04-08T21:13:28.257330Z
Exterior Derivative of a $C^1$ Differential Form on a Euclidean Open Set
definitiondef:exterior-derivative-c1-differential-form-euclidean-open-set-2026bGeometry Multivariable Calculus
Let $n\in\mathbb{N}$ , let $U\subseteq \mathbb{R}^n$ be open, let $k\in\mathbb{N}\cup\{0\}$ , and let $\omega$ be a $C^1$ differential $k$ -form on $U$ . Write $\omega=\sum_{1\le i_1<\cdots<i_k\le n} a_{i_1\dots i_k}\, dx_{i_1}\wedge\cdots\wedge dx_{i_k}$ as in…
+0 / -0flags 0verified 0no proof
Authors ChatGPT-5.4, Aaron · Created 2026-04-08T21:13:28.229940Z
$C^1$ Differential k-Form on an Open Subset of Euclidean Space
definitiondef:c1-differential-k-form-euclidean-open-set-2026bGeometry Multivariable Calculus
Let $n\in\mathbb{N}$ , let $U\subseteq \mathbb{R}^n$ be open, let $k\in\mathbb{N}\cup\{0\}$ , and let $\omega$ be a differential $k$ -form on $U$ . Write $\omega$ in the coordinate expansion from Coordinate Expansion of Differential Forms on Euclidean Open Sets. We say that $\omega$ …
+0 / -0flags 0verified 0no proof
Authors ChatGPT-5.4, Aaron · Created 2026-04-08T21:13:28.196297Z
Continuous Differential k-Form on an Open Subset of Euclidean Space
definitiondef:continuous-differential-k-form-euclidean-open-set-2026bGeometry Multivariable Calculus
Let $n\in\mathbb{N}$ , let $U\subseteq \mathbb{R}^n$ be open, let $k\in\mathbb{N}\cup\{0\}$ , and let $\omega$ be a differential $k$ -form on $U$ . Write $\omega$ in the coordinate expansion from Coordinate Expansion of Differential Forms on Euclidean Open Sets. We say that $\omega$ …
+0 / -0flags 0verified 0no proof
Authors ChatGPT-5.4, Aaron · Created 2026-04-08T21:13:28.159322Z
Coordinate Expansion of Differential Forms on Euclidean Open Sets
theoremthm:coordinate-expansion-differential-forms-euclidean-2026bGeometry Multivariable Calculus
Let $n\in\mathbb{N}$ , let $U\subseteq \mathbb{R}^n$ be open, let $k\in\mathbb{N}\cup\{0\}$ , and let $\omega$ be a differential $k$ -form on $U$ . Then there exist unique real-valued functions $a_{i_1\dots i_k}:U\to\mathbb{R}$ indexed by strictly increasing $k$ -tuples…
+0 / -0flags 0verified 0no proof
Authors ChatGPT-5.4, Aaron · Created 2026-04-08T21:13:28.124906Z
Coordinate 1-Form on an Open Subset of Euclidean Space
definitiondef:coordinate-1-form-euclidean-open-set-2026aGeometry Multivariable Calculus
Let $n\in\mathbb{N}$ , let $U\subseteq \mathbb{R}^n$ be open, and let $i\in\{1,\dots,n\}$ . The $i$ th coordinate $1$ -form on $U$ is the differential $1$ -form $dx_i$ on $U$ defined by $(dx_i)_x(v)=v_i$ for every point $x\in U$ and every vector…
+0 / -0flags 0verified 0no proof
Authors ChatGPT-5.4, Aaron · Created 2026-04-08T21:13:28.050001Z
Composition of Continuous Euclidean Maps
theoremthm:composition-continuous-euclidean-2026aMultivariable Calculus
Let $n,m,p\in\mathbb{N}$ . Let $E\subseteq \mathbb{R}^n$ , let $F\subseteq \mathbb{R}^m$ , let $f:E\to F$ , and let $g:F\to\mathbb{R}^p$ . Let $a\in E$ . Suppose that $f$ is continuous at $a$ and that $g$ is continuous at $f(a)$ . Then the composition $g\circ f:E\to\mathbb{R}^p$ is cont…
+1 / -0flags 0verified 2has proof
Authors ChatGPT-5.4, Aaron · Created 2026-04-06T21:14:19.915854Z
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. 1. The map $f$ is continuous at $a$ . 2. For every index $j\in\{1,\dots,m\}$ , the coordinate function $f_j:E\to\mathbb{R}$ is cont…
+1 / -0flags 0verified 2has proof
Authors ChatGPT-5.4, Aaron · Created 2026-04-06T21:14:19.736365Z
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.
+0 / -0flags 0verified 0no proof
Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:27:09.013170Z
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 open subsets. Let $f:U\to V$ and $g:V\to W$ be $C^1$ maps. Then the composition $g\circ f:U\to W$ is again of class $C^1$ . Moreover, for every $a\in U$ , the Jacob…
+1 / -0flags 0verified 2has proof
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-2026aGeometry Multivariable Calculus
Let $n,m,k\in\mathbb{N}$ . Let $U\subseteq \mathbb{R}^n$ and $V\subseteq \mathbb{R}^m$ be open subsets, let $F:U\to V$ be a $C^1$ map, and let $\omega$ be a differential $k$ -form on $V$ . The pullback of $\omega$ by $F$ is the differential $k$ -form $F^{*}\omega$ on $U$ defined by…
+0 / -0flags 0verified 0no proof
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-2026bGeometry Multivariable Calculus
Let $n\in\mathbb{N}$ and let $k,\ell\in\mathbb{N}\cup\{0\}$ . Let $U\subseteq \mathbb{R}^n$ be open. Let $\alpha$ be a 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…
+0 / -0flags 0verified 0no proof
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-2026aGeometry Multivariable Calculus
Let $n\in\mathbb{N}$ , let $U\subseteq \mathbb{R}^n$ be 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 alternating $k$ -linear form $\omega_x$ on $\mathbb{R}^n$ . For vectors…
+0 / -0flags 0verified 0no proof
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-2026aGeometry Multivariable 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…
+0 / -1flags 0verified 0no proof
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 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 continuous at every point of $U$ , and if for every $j\in\{1,\dots,m\}$ and eve…
+0 / -1flags 0verified 0no proof
Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:02:45.653161Z
Continuity at a Point for Maps Between Euclidean Spaces
definitiondef:continuous-map-at-point-euclidean-2026aMultivariable Calculus
Let $n,m\in\mathbb{N}$ . Let $E\subseteq \mathbb{R}^n$ , let $f:E\to\mathbb{R}^m$ , let $a\in E$ , and write $f=(f_1,\dots,f_m)$ . We say that $f$ is continuous at $a$ if for every $\varepsilon>0$ there exists $\delta>0$ such that for every point $x=(x_1,\dots,x_n)\in E$ , if…
+0 / -0flags 0verified 0no proof
Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:02:45.446840Z
Open Subset of Euclidean Space
definitiondef:open-subset-euclidean-space-2026aMultivariable Calculus
Let $n\in\mathbb{N}$ and let $U\subseteq \mathbb{R}^n$ . We say that $U$ is open in $\mathbb{R}^n$ if for every point $x=(x_1,\dots,x_n)\in U$ there exists a real number $r>0$ such that every point $y=(y_1,\dots,y_n)\in\mathbb{R}^n$ satisfying $\sum_{i=1}^n (y_i-x_i)^2<r^2$ al…
+0 / -0flags 0verified 0no proof
Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:02:45.184360Z

Showing 181-199 of 199

Back

Integral of a Differential Form over an Oriented k-Sub-Rectangle in Euclidean Space

Boundary of an Oriented k-Sub-Rectangle in Euclidean Space

Standard k-Rectangle in Euclidean Space

Exterior Derivative of a C1C^1C1 Differential Form on a Euclidean Open Set

C1C^1C1 Differential k-Form on an Open Subset of Euclidean Space

Continuous Differential k-Form on an Open Subset of Euclidean Space

Coordinate Expansion of Differential Forms on Euclidean Open Sets

Coordinate 1-Form on an Open Subset of Euclidean Space

Composition of Continuous Euclidean Maps

Coordinatewise Characterization of Continuity for Euclidean Maps

Euclidean Space Rn\mathbb{R}^nRn

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

Continuity at a Point for Maps Between Euclidean Spaces

Open Subset of Euclidean Space

Exterior Derivative of a $C^1$ Differential Form on a Euclidean Open Set

$C^1$ Differential k-Form on an Open Subset of Euclidean Space

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