Theorems

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

Showing 161-180 of 199

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Interior Points, Boundary Points, and Boundary of a Smooth Manifold with Boundary
definitiondef:boundary-smooth-manifold-with-boundary-2026aTopology Geometry Multivariable Calculus
Let $M$ be a smooth manifold with boundary of dimension $n$ in the sense of Smooth Atlas and Smooth Manifold with Boundary, and let $p\in M$ . We say that $p$ is an interior point of $M$ if there exists a chart $(U,\varphi)$ in the chosen atlas with $p\in U$ and…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T21:58:23.620708Z
Smooth Atlas and Smooth Manifold with Boundary
definitiondef:smooth-manifold-with-boundary-2026aTopology Geometry 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-2026aTopology Geometry 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-2026aTopology Geometry 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-2026aTopology Geometry 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
Heine-Borel Theorem in $\mathbb{R}^n$
theoremthm:heine-borel-rn-2026aAnalysis Topology Multivariable Calculus
Let $n\in\mathbb{N}$ , and let $A\subseteq\mathbb{R}^n$ . Then the following are equivalent. 1. $A$ is compact in $\mathbb{R}^n$ , where $\mathbb{R}^n$ is regarded as a topological space through the topology determined by the Euclidean distance. 2. $A$ is closed in $\mathbb{R}^n$ a…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-10T00:21:46.209722Z
Compact Subset of $\mathbb{R}^n$ is Closed
theoremthm:compact-subset-rn-closed-2026aAnalysis Topology Multivariable Calculus
Let $n\in\mathbb{N}$ , and let $A\subseteq\mathbb{R}^n$ . Assume that $A$ is compact in $\mathbb{R}^n$ , where $\mathbb{R}^n$ is regarded as a topological space through the topology determined by the Euclidean distance. Then $A$ is closed in $\mathbb{R}^n$ .
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Authors ChatGPT-5.4, Aaron · Created 2026-04-10T00:21:46.154631Z
Compact Subset of $\mathbb{R}^n$ is Bounded
theoremthm:compact-subset-rn-bounded-2026aAnalysis Topology Multivariable Calculus
Let $n\in\mathbb{N}$ , and let $A\subseteq\mathbb{R}^n$ . Assume that $A$ is compact in $\mathbb{R}^n$ , where $\mathbb{R}^n$ is regarded as a topological space through the topology determined by the Euclidean distance. Then $A$ is bounded as a subset of the metric space…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-10T00:21:46.110472Z
Closed Box in $\mathbb{R}^n$ is Compact
theoremthm:closed-box-compact-rn-2026aAnalysis Topology Multivariable Calculus
Let $n\in\mathbb{N}$ . For each index $i\in\{1,\dots,n\}$ , let $a_i,b_i\in\mathbb{R}$ satisfy $a_i\le b_i$ , and let $B\subseteq\mathbb{R}^n$ be the closed box determined by these endpoints. Then $B$ is compact in $\mathbb{R}^n$ , where $\mathbb{R}^n$ is regarded as a topological sp…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-10T00:21:46.055462Z
Euclidean Open Box Criterion in $\mathbb{R}^n$
theoremthm:euclidean-open-box-criterion-rn-2026aTopology Multivariable Calculus
Let $n\in\mathbb{N}$ and let $U\subseteq \mathbb{R}^n$ . Then $U$ is open in the Euclidean sense if and only if for every point $x=(x_1,\dots,x_n)\in U$ there exists a real number $\delta>0$ such that every point $y=(y_1,\dots,y_n)\in\mathbb{R}^n$ satisfying…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-10T00:21:46.001092Z
Closed Box in $\mathbb{R}^n$
definitiondef:closed-box-rn-2026aTopology Multivariable Calculus
Let $n\in\mathbb{N}$ . For each index $i\in\{1,\dots,n\}$ , let $a_i,b_i\in\mathbb{R}$ satisfy $a_i\le b_i$ . The subset $B=\{x=(x_1,\dots,x_n)\in\mathbb{R}^n : a_i\le x_i\le b_i \text{ for every } i\in\{1,\dots,n\}\}$ is called the closed box in $\mathbb{R}^n$ determined by the…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-10T00:21:45.833615Z
Euclidean Openness Agrees with Metric Openness on $\mathbb{R}^n$
theoremthm:euclidean-open-iff-metric-open-rn-2026aTopology Multivariable Calculus
Let $n\in\mathbb{N}$ , let $U\subseteq \mathbb{R}^n$ , and let $d_E$ be the Euclidean distance on $\mathbb{R}^n$ . Then $U$ is open in the Euclidean sense if and only if $U$ is open in the metric space $(\mathbb{R}^n,d_E)$ .
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Authors ChatGPT-5.4, Aaron · Created 2026-04-09T23:53:24.013485Z
Euclidean Distance is a Metric on $\mathbb{R}^n$
theoremthm:euclidean-distance-is-metric-rn-2026aTopology Multivariable Calculus
Let $n\in\mathbb{N}$ . Then the Euclidean distance $d_E$ is a metric on $\mathbb{R}^n$ .
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Authors ChatGPT-5.4, Aaron · Created 2026-04-09T23:53:23.974349Z
Euclidean Distance on $\mathbb{R}^n$
definitiondef:euclidean-distance-rn-2026aTopology Multivariable Calculus
Let $n\in\mathbb{N}$ . Consider the set $\mathbb{R}^n$ appearing in the definition of open subsets of Euclidean space. For points $x=(x_1,\dots,x_n),\qquad y=(y_1,\dots,y_n)$ in $\mathbb{R}^n$ , define $d_E(x,y)=\sqrt{\sum_{i=1}^n (x_i-y_i)^2},$ where $\sqrt{\ \cdot\ }$ den…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-09T23:53:23.927597Z
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
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…
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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…
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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

Showing 161-180 of 199

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Interior Points, Boundary Points, and Boundary of a Smooth Manifold with Boundary

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

Heine-Borel Theorem in Rn\mathbb{R}^nRn

Compact Subset of Rn\mathbb{R}^nRn is Closed

Compact Subset of Rn\mathbb{R}^nRn is Bounded

Closed Box in Rn\mathbb{R}^nRn is Compact

Euclidean Open Box Criterion in Rn\mathbb{R}^nRn

Closed Box in Rn\mathbb{R}^nRn

Euclidean Openness Agrees with Metric Openness on Rn\mathbb{R}^nRn

Euclidean Distance is a Metric on Rn\mathbb{R}^nRn

Euclidean Distance on Rn\mathbb{R}^nRn

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

Heine-Borel Theorem in $\mathbb{R}^n$

Compact Subset of $\mathbb{R}^n$ is Closed

Compact Subset of $\mathbb{R}^n$ is Bounded

Closed Box in $\mathbb{R}^n$ is Compact

Euclidean Open Box Criterion in $\mathbb{R}^n$

Closed Box in $\mathbb{R}^n$

Euclidean Openness Agrees with Metric Openness on $\mathbb{R}^n$

Euclidean Distance is a Metric on $\mathbb{R}^n$

Euclidean Distance on $\mathbb{R}^n$

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