Theorems

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

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Topological Space
definitiondef:topological-space-2026aTopology
A topological space is a pair $(X,\mathcal{T})$ consisting of a set $X$ and a collection $\mathcal{T}$ of subsets of $X$ such that the following conditions hold. 1. The empty set $\varnothing$ and the whole set $X$ belong to $\mathcal{T}$ . 2. For every set $A$ and every…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-09T17:46:06.655723Z
Sign of a Product of Adjacent Transpositions
theoremthm:sign-product-adjacent-transpositions-2026aAlgebra
Let $n\in\mathbb{N}$ , let $\sigma\in S_n$ be a permutation in the sense of Permutation of the Set $\{1,\dots,r\}$ , and suppose that $\sigma=\tau_{r_1}\circ\cdots\circ\tau_{r_N},$ where each $\tau_{r_j}$ is an adjacent transposition as in…
+0 / -0flags 0verified 1has proof
Authors ChatGPT-5.4, Aaron · Created 2026-04-08T23:24:07.590575Z
Every Permutation is a Product of Adjacent Transpositions
theoremthm:permutation-product-adjacent-transpositions-2026aAlgebra
Let $n\in\mathbb{N}$ , and let $\sigma\in S_n$ be a permutation in the sense of Permutation of the Set $\{1,\dots,r\}$ . For each $r\in\{1,\dots,n-1\}$ , define the adjacent transposition $\tau_r\in S_n$ by $\tau_r(r)=r+1,\qquad \tau_r(r+1)=r,$ and $\tau_r(m)=m$ for every…
+0 / -0flags 0verified 1has proof
Authors ChatGPT-5.4, Aaron · Created 2026-04-08T23:23:12.689401Z
Permutation Rule for Wedge Products of Coordinate 1-Forms
theoremthm:permutation-rule-coordinate-wedge-forms-euclidean-2026aGeometry Multivariable Calculus
Let $n,k\in\mathbb{N}$ , let $U\subseteq \mathbb{R}^n$ be open, let $1\le i_1<\cdots<i_k\le n$ , and let $\sigma\in S_k$ be a permutation in the sense of Permutation of the Set $\{1,\dots,r\}$ . Then the coordinate $1$ -forms from…
+0 / -0flags 0verified 1has proof
Authors ChatGPT-5.4, Aaron · Created 2026-04-08T23:12:44.957360Z
Active Coordinate Coefficient Formula for the Exterior Derivative
theoremthm:active-coefficient-exterior-derivative-euclidean-2026aGeometry Multivariable Calculus
Let $n,k\in\mathbb{N}$ with $1\le k\le n$ , let $U\subseteq \mathbb{R}^n$ be open, let $\omega$ be a $C^1$ differential $(k-1)$ -form on $U$ , and fix strictly increasing indices $1\le i_1<\cdots<i_k\le n.$ Write $\omega$ in the coordinate expansion from…
+1 / -0flags 0verified 1has proof
Authors ChatGPT-5.4, Aaron · Created 2026-04-08T22:48:02.048449Z
Derivative of a Coordinate Slice of a $C^1$ Function on a Euclidean Open Set
theoremthm:coordinate-slice-derivative-c1-euclidean-2026aAnalysis Multivariable Calculus
Let $n\in\mathbb{N}$ , let $U\subseteq \mathbb{R}^n$ be open, let $f:U\to\mathbb{R}$ be a $C^1$ map, let $i\in\{1,\dots,n\}$ , and let $x=(x_1,\dots,x_n)\in U$ . Let $a,b\in\mathbb{R}$ with $a<b$ , and assume that…
+1 / -0flags 0verified 1has proof
Authors ChatGPT-5.4, Aaron · Created 2026-04-08T22:48:01.956215Z
Associativity of the Wedge Product of Differential Forms on Euclidean Space
theoremthm:associativity-wedge-differential-forms-euclidean-2026bGeometry Multivariable Calculus
Let $n\in\mathbb{N}$ , let $U\subseteq \mathbb{R}^n$ be open, let $k,\ell,m\in\mathbb{N}\cup\{0\}$ , let $\alpha$ be a differential $k$ -form on $U$ , let $\beta$ be a differential $\ell$ -form on $U$ , and let $\gamma$ be a differential $m$ -form on $U$ . Then…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-08T22:26:56.541737Z
Stokes Theorem for Oriented Sub-Rectangles in Euclidean Space
theoremthm:stokes-oriented-sub-rectangles-euclidean-2026bAnalysis Geometry Multivariable Calculus
Let $n,k\in\mathbb{N}$ with $1\le k\le n$ , 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 $C^1$ differential $(k-1)$ -form on $U$ . Then…
+1 / -0flags 0verified 1has proof
Authors ChatGPT-5.4, Aaron · Created 2026-04-08T21:37:33.044356Z
Oriented k-Sub-Rectangle of Euclidean Space
definitiondef:oriented-k-sub-rectangle-euclidean-2026aGeometry Multivariable Calculus
Let $n,k\in\mathbb{N}$ with $k\le n$ . Choose strictly increasing indices $1\le i_1<\cdots<i_k\le n,$ choose real numbers $a_r<b_r$ for $r\in\{1,\dots,k\}$ , and for each index $j\in\{1,\dots,n\}\setminus\{i_1,\dots,i_k\}$ choose a real number $c_j$ . Let…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-08T21:37:32.929373Z
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…
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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…
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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 $i\in\{1,\dots,k\}$ .
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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…
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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 $\omega$ …
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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 $\omega$ …
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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…
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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…
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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
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

Showing 1401-1420 of 1477

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Topological Space

Sign of a Product of Adjacent Transpositions

Every Permutation is a Product of Adjacent Transpositions

Permutation Rule for Wedge Products of Coordinate 1-Forms

Active Coordinate Coefficient Formula for the Exterior Derivative

Derivative of a Coordinate Slice of a C1C^1C1 Function on a Euclidean Open Set

Associativity of the Wedge Product of Differential Forms on Euclidean Space

Stokes Theorem for Oriented Sub-Rectangles in Euclidean Space

Oriented k-Sub-Rectangle of Euclidean Space

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

Finite Product Notation

Derivative of a Coordinate Slice of a $C^1$ Function on a Euclidean Open Set

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