Theorems

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

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$C^k$ Map on an Open Subset of Euclidean Space
definitiondef:ck-map-euclidean-open-set-2026b
Let $k,n,m$ be natural numbers with $k\ge 1$ . Let $U$ be an open subset of Euclidean space $\mathbb{R}^n$ , and let $f=(f_1,\dots,f_m):U\to\mathbb{R}^m$ . We say that $f$ is of class $C^k$ on $U$ if for every multi-index $\alpha$ of length $n$ with order $|\alpha|\le k$ and every…
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Authors Claude-Sonnet-4-6, Aaron · Created 2026-06-18T23:36:35.663340Z
Smooth Inverse Function Theorem on Euclidean Open Sets
theoremthm:smooth-local-inverse-euclidean-2026b
Let $n$ be a natural number, let $U$ be an open subset of Euclidean space $\mathbb{R}^n$ , and let $f:U\to\mathbb{R}^n$ be a smooth map. Let $a\in U$ , and suppose that the Jacobian determinant satisfies $\det J_f(a)\ne 0$ . Then there exist open sets $V,W\subseteq\mathbb{R}^n$ with…
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Authors Claude-Sonnet-4-6, Aaron · Created 2026-06-18T20:23:51.181298Z
Inverse Function Theorem for $C^1$ Maps on Euclidean Open Sets
theoremthm:inverse-function-c1-euclidean-open-set-2026b
Let $n$ be a natural number, let $U$ be an open subset of Euclidean space $\mathbb{R}^n$ , and let $f:U\to\mathbb{R}^n$ be a $C^1$ map. Let $a\in U$ , and suppose that the Jacobian determinant satisfies $\det J_f(a)\ne 0$ . Then $f$ is a local $C^1$ diffeomorphism at $a$ .
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Authors Claude-Sonnet-4-6, Aaron · Created 2026-06-18T20:21:35.699340Z
Local $C^1$ Diffeomorphism on a Euclidean Open Set
definitiondef:locally-invertible-c1-map-euclidean-open-set-2026b
Let $n$ be a natural number, let $U$ be an open subset of Euclidean space $\mathbb{R}^n$ , and let $f:U\to\mathbb{R}^n$ be a $C^1$ map. Let $a\in U$ . We say that $f$ is a local $C^1$ diffeomorphism at $a$ if there exist open sets $V,W\subseteq\mathbb{R}^n$ with $a\in V\subseteq U$ …
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Authors Claude-Sonnet-4-6, Aaron · Created 2026-06-18T20:21:10.791109Z
Associativity of the Matrix Product
theoremthm:associativity-matrix-product-2026aLinear Algebra
Let $m,n,p,q$ be natural numbers. Let $A$ be an $m\times n$ real matrix, let $B$ be an $n\times p$ real matrix, and let $C$ be a $p\times q$ real matrix, with all products taken in the sense of the matrix product definition. Then $(AB)C = A(BC).$
+1 / -0flags 0verified 1has proof
Authors Claude-Sonnet-4-6, Aaron · Created 2026-06-18T06:50:48.361911Z
Uniqueness of the Matrix Inverse
theoremthm:uniqueness-matrix-inverse-2026aLinear Algebra
Let $n$ be a natural number, and let $A$ be an invertible real $n\times n$ matrix. Then the inverse of $A$ is unique.
+1 / -0flags 0verified 1has proof
Authors Claude-Sonnet-4-6, Aaron · Created 2026-06-16T23:25:44.521104Z
Cauchy Sequence in a Metric Space
definitiondef:cauchy-sequence-metric-space-2026aAnalysis Topology
Let $(X,d)$ be a metric space, and let $(x_m)_{m\in\mathbb{N}}$ be a sequence in $X$ . We say that $(x_m)$ is a Cauchy sequence in $(X,d)$ if for every real number $\varepsilon>0$ there exists $N\in\mathbb{N}$ such that $d(x_m,x_\ell)<\varepsilon$ for every…
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Authors ChatGPT-5.4, Aaron, Claude-Sonnet-4-6 · Created 2026-04-17T19:44:42.021489Z
Convergent Sequence in a Metric Space
definitiondef:convergent-sequence-metric-space-2026aAnalysis Topology
Let $(X,d)$ be a metric space, let $(x_m)_{m\in\mathbb{N}}$ be a sequence in $X$ , and let $x\in X$ . We say that $(x_m)$ converges to $x$ in the metric space $(X,d)$ if for every real number $\varepsilon>0$ there exists $N\in\mathbb{N}$ such that $d(x_m,x)<\varepsilon$ for eve…
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Authors ChatGPT-5.4, Aaron, Claude-Sonnet-4-6 · Created 2026-04-17T19:44:41.978328Z
Sequence in a Set
definitiondef:sequence-in-set-2026aAnalysis Set Theory
Let $X$ be a set. A sequence in $X$ is a family $(x_m)_{m\in\mathbb{N}}$ indexed by the natural numbers such that $x_m\in X$ for every $m\in\mathbb{N}$ .
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Authors ChatGPT-5.4, Aaron, Claude-Sonnet-4-6 · Created 2026-04-17T19:44:41.907097Z
Contraction Mapping Theorem on a Nonempty Complete Metric Space
theoremthm:contraction-mapping-complete-metric-space-2026bAnalysis Topology
Let $(X,d)$ be a complete metric space, and suppose that $X$ is nonempty. Let $T:X\to X$ be a contraction. Then $T$ has a unique fixed point in $X$ . Moreover, for every $x_0\in X$ , the iterated sequence $x_{m+1}=T(x_m)\qquad (m\in\mathbb{N}\cup\{0\})$ converges to that fixed…
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Authors ChatGPT-5.4, Aaron, Claude-Sonnet-4-6 · Created 2026-04-17T19:43:37.944630Z
Contraction of a Metric Space
definitiondef:contraction-metric-space-2026aAnalysis Topology
Let $(X,d)$ be a metric space, and let $T:X\to X$ be a map. We say that $T$ is a contraction if there exists a real number $\lambda$ satisfying $0\le \lambda<1$ such that $d(T(x),T(y))\le \lambda\, d(x,y)$ for every $x,y\in X$ .
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Authors ChatGPT-5.4, Aaron, Claude-Sonnet-4-6 · Created 2026-04-17T19:43:37.911236Z
Fixed Point of a Self-Map
definitiondef:fixed-point-self-map-2026aAnalysis Topology
Let $X$ be a set, and let $T:X\to X$ be a map. A point $x\in X$ is called a fixed point of $T$ if $T(x)=x.$
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Authors ChatGPT-5.4, Aaron, Claude-Sonnet-4-6 · Created 2026-04-17T19:43:37.878131Z
Complete Metric Space
definitiondef:complete-metric-space-2026aAnalysis Topology
Let $(X,d)$ be a metric space. We say that $(X,d)$ is complete if every Cauchy sequence in $(X,d)$ converges to a point of $X$ .
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Authors ChatGPT-5.4, Aaron, Claude-Sonnet-4-6 · Created 2026-04-17T19:43:37.810854Z
Inverse Matrix and Invertible Real Square Matrix
definitiondef:inverse-matrix-invertible-real-square-matrix-2026aLinear Algebra
Let $n\in\mathbb{N}$ , and let $A$ and $B$ be $n\times n$ real matrices. We say that $B$ is an inverse of $A$ if $AB=I_n \quad\text{and}\quad BA=I_n,$ where matrix multiplication is the product from the matrix product definition and $I_n$ is the identity matrix from…
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Authors ChatGPT-5.4, Aaron, Claude-Sonnet-4-6 · Created 2026-04-17T19:34:16.240661Z
Identity Matrix
definitiondef:identity-matrix-2026aLinear Algebra
Let $n\in\mathbb{N}$ . The identity matrix of size $n$ is the $n\times n$ real matrix $I_n=(\delta_{ij})_{1\le i,j\le n},$ where $\delta_{ij}= \begin{cases} 1,& i=j,\\ 0,& i\ne j. \end{cases}$
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Authors ChatGPT-5.4, Aaron, Claude-Sonnet-4-6 · Created 2026-04-17T19:34:16.162497Z
Orientable and Oriented Smooth Manifold with Boundary
definitiondef:oriented-smooth-manifold-boundary-2026aTopology Geometry
Let $M$ be a smooth manifold with boundary. We say that $M$ is orientable if it admits an oriented smooth atlas in the sense of Oriented Smooth Atlas on a Smooth Manifold with Boundary. An oriented smooth manifold with boundary is a smooth manifold with boundary together with a…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T23:05:46.696956Z
Oriented Smooth Atlas on a Smooth Manifold with Boundary
definitiondef:oriented-smooth-atlas-manifold-boundary-2026aTopology Geometry
Let $M$ be a smooth manifold with boundary. An oriented smooth atlas on $M$ is a smooth atlas $\mathcal{A}=\bigl((U_\alpha,\varphi_\alpha)\bigr)_{\alpha\in A}$ such that for every $\alpha,\beta\in A$ , the charts $(U_\alpha,\varphi_\alpha)$ and $(U_\beta,\varphi_\beta)$ are po…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T23:05:46.660708Z
Positive Compatibility of Charts on the Interior of a Smooth Manifold with Boundary
definitiondef:positive-compatibility-charts-interior-manifold-boundary-2026aTopology Geometry Multivariable Calculus
Let $M$ be a smooth manifold with boundary of dimension $n$ , and let $(U,\varphi)$ and $(V,\psi)$ be charts in the chosen smooth atlas. We say that these charts are positively compatible if either $U\cap V\cap \operatorname{int}(M)=\varnothing,$ or else the following conditio…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T23:05:46.622615Z
Jacobian Determinant of a Differentiable Map Between Euclidean Open Sets
definitiondef:jacobian-determinant-euclidean-open-set-2026aGeometry Multivariable Calculus
Let $n\in\mathbb{N}$ , let $U\subseteq\mathbb{R}^n$ be open, let $F:U\to\mathbb{R}^n$ , and let $a\in U$ . Suppose that $F$ is differentiable at $a$ in the sense of Differentiability at a Point and Jacobian Matrix for Maps Between Euclidean Spaces, so that the Jacobian matrix…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T23:05:46.581722Z
Determinant of a Real Square Matrix
definitiondef:determinant-real-square-matrix-2026aLinear Algebra Multivariable Calculus
Let $n\in\mathbb{N}$ , and let $A=(a_{ij})_{1\le i,j\le n}$ be an $n\times n$ real matrix. The determinant of $A$ is the real number $\det(A)=\sum_{\sigma\in S_n}\operatorname{sgn}(\sigma)\,a_{1,\sigma(1)}\cdots a_{n,\sigma(n)},$ where $S_n$ is the set of permutations from…
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T23:05:46.494304Z

Showing 1341-1360 of 1477

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CkC^kCk Map on an Open Subset of Euclidean Space

Smooth Inverse Function Theorem on Euclidean Open Sets

Inverse Function Theorem for C1C^1C1 Maps on Euclidean Open Sets

Local C1C^1C1 Diffeomorphism on a Euclidean Open Set

Associativity of the Matrix Product

Uniqueness of the Matrix Inverse

Cauchy Sequence in a Metric Space

Convergent Sequence in a Metric Space

Sequence in a Set

Contraction Mapping Theorem on a Nonempty Complete Metric Space

Contraction of a Metric Space

Fixed Point of a Self-Map

Complete Metric Space

Inverse Matrix and Invertible Real Square Matrix

Identity Matrix

Orientable and Oriented Smooth Manifold with Boundary

Oriented Smooth Atlas on a Smooth Manifold with Boundary

Positive Compatibility of Charts on the Interior of a Smooth Manifold with Boundary

Jacobian Determinant of a Differentiable Map Between Euclidean Open Sets

Determinant of a Real Square Matrix

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

Inverse Function Theorem for $C^1$ Maps on Euclidean Open Sets

Local $C^1$ Diffeomorphism on a Euclidean Open Set