Theorems

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

Showing 21-40 of 41

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Smooth Atlas and Smooth Manifold with Boundary
definitiondef:smooth-manifold-with-boundary-2026aGeometry Topology Multivariable Calculus
Let $(X,\mathcal{T})$ be a topological space, and let $n\in\mathbb{N}$ . A smooth atlas of dimension $n$ on $X$ , modeled on the closed upper half-space, is a family of charts $\mathcal{A}=\bigl((U_\alpha,\varphi_\alpha)\bigr)_{\alpha\in A}$ for some set $A$ such that the follo…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T21:58:23.568526Z
Smooth Compatibility of Charts Modeled on the Closed Upper Half-Space
definitiondef:smooth-compatible-charts-upper-half-space-2026aGeometry Topology Multivariable Calculus
Let $(X,\mathcal{T})$ be a topological space, let $n\in\mathbb{N}$ , and let $(U,\varphi)$ and $(V,\psi)$ be charts of dimension $n$ on $X$ in the sense of Chart Modeled on the Closed Upper Half-Space. Write…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T21:58:23.517394Z
Chart Modeled on the Closed Upper Half-Space
definitiondef:chart-upper-half-space-2026aGeometry Topology Multivariable Calculus
Let $(X,\mathcal{T})$ be a topological space, let $n\in\mathbb{N}$ , and let $U\subseteq X$ . A chart of dimension $n$ on $X$ , modeled on the closed upper half-space, is a pair $(U,\varphi)$ with the following properties. 1. $U\in\mathcal{T}$ . 2. If $H^n$ denotes the half-space fr…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T21:58:23.439943Z
Closed Upper Half-Space in Euclidean Space
definitiondef:closed-upper-half-space-euclidean-2026aGeometry Topology Multivariable Calculus
Let $n\in\mathbb{N}$ . In the Euclidean space $\mathbb{R}^n$ , the closed upper half-space is the subset $H^n=\{x=(x_1,\dots,x_n)\in\mathbb{R}^n : x_n\ge 0\}.$ A subset $\Omega\subseteq H^n$ is said to be open in $H^n$ if there exists an open subset $U\subseteq\mathbb{R}^n$ suc…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T21:58:23.308760Z
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…
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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…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-08T22:48:02.048449Z
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…
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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
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…
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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-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…
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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-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…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T20:02:45.729237Z

Showing 21-40 of 41

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Smooth Atlas and Smooth Manifold with Boundary

Smooth Compatibility of Charts Modeled on the Closed Upper Half-Space

Chart Modeled on the Closed Upper Half-Space

Closed Upper Half-Space in Euclidean Space

Permutation Rule for Wedge Products of Coordinate 1-Forms

Active Coordinate Coefficient Formula for the Exterior Derivative

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

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

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

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