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Smooth Inverse Function Theorem on Euclidean Open Sets
theoremthm:smooth-local-inverse-euclidean-2026b
Let $n$ be a \reftext{def:natural-numbers-2026a}{natural number}, let $U$ be an \reftext{def:open-subset-euclidean-space-2026a}{open} subset of \reftext{def:euclidean-space-rn-2026a}{Euclidean space} $\mathbb{R}^n$ , and let $f:U\to\mathbb{R}^n$ be a \reftext{def:smooth-map-euclidean-open-set-2026a}{smooth} map. Let $a\in U$ , and suppose that the \reftext{def:jacobian-determinant-euclidean-open-set-2026a}{Jacobian determinant} satisfies $\det J_f(a)\ne 0$ . Then there exist \reftext{def:open-subset-euclidean-space-2026a}{open} sets $V,W\subseteq\mathbb{R}^n$ with $a\in V\subseteq U$ and $f(a)\in W$ such that $f(V)=W$ , the restriction $f|_V:V\to W$ is \reftext{def:bijection-sets-2026a}{bijective}, and its inverse $h=(f|_V)^{-1}:W\to V$ is \reftext{def:smooth-map-euclidean-open-set-2026a}{smooth}. Moreover, the \reftext{def:differentiable-map-at-point-euclidean-2026a}{Jacobian matrix} of $h$ satisfies $J_h(y)=\bigl(J_f(h(y))\bigr)^{-1}$ for all $y\in W$ , where $(\cdot)^{-1}$ denotes the \reftext{def:inverse-matrix-invertible-real-square-matrix-2026a}{matrix inverse}.
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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 \reftext{def:natural-numbers-2026a}{natural number}, let $U$ be an \reftext{def:open-subset-euclidean-space-2026a}{open} subset of \reftext{def:euclidean-space-rn-2026a}{Euclidean space} $\mathbb{R}^n$ , and let $f:U\to\mathbb{R}^n$ be a \reftext{def:c1-map-euclidean-open-set-2026a}{ $C^1$ map}. Let $a\in U$ , and suppose that the \reftext{def:jacobian-determinant-euclidean-open-set-2026a}{Jacobian determinant} satisfies $\det J_f(a)\ne 0$ . Then $f$ is a \reftext{def:locally-invertible-c1-map-euclidean-open-set-2026a}{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 \reftext{def:natural-numbers-2026a}{natural number}, let $U$ be an \reftext{def:open-subset-euclidean-space-2026a}{open} subset of \reftext{def:euclidean-space-rn-2026a}{Euclidean space} $\mathbb{R}^n$ , and let $f:U\to\mathbb{R}^n$ be a \reftext{def:c1-map-euclidean-open-set-2026a}{ $C^1$ map}. Let $a\in U$ . We say that $f$ is a \textit{local $C^1$ diffeomorphism at $a$ } if there exist \reftext{def:open-subset-euclidean-space-2026a}{open} sets $V,W\subseteq\mathbb{R}^n$ with $a\in V\subseteq U$ and $f(a)\in W$ such that $f(V)=W$ , the restriction $f|_V:V\to W$ is \reftext{def:bijection-sets-2026a}{bijective}, and its inverse $(f|_V)^{-1}:W\to V$ is \reftext{def:c1-map-euclidean-open-set-2026a}{of class $C^1$ }. We say $f$ is a \textit{local $C^1$ diffeomorphism} (on $U$ ) if $f$ is a local $C^1$ diffeomorphism at every $a\in U$ .
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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 \reftext{def:natural-numbers-2026a}{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 \reftext{def:product-real-matrices-2026a}{the matrix product definition}. Then $(AB)C = A(BC).$
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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 \reftext{def:natural-numbers-2026a}{natural number}, and let $A$ be an \reftext{def:inverse-matrix-invertible-real-square-matrix-2026a}{invertible} real $n\times n$ matrix. Then the inverse of $A$ is unique.
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Authors Claude-Sonnet-4-6, Aaron · Created 2026-06-16T23:25:44.521104Z
Cauchy Sequence in a Metric Space
definitiondef:cauchy-sequence-metric-space-2026aAnalysisTopology
Let $(X,d)$ be a \reftext{def:metric-space-2026a}{metric space}, and let $(x_m)_{m\in\mathbb{N}}$ be a \reftext{def:sequence-in-set-2026a}{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 $m,\ell\in\mathbb{N}$ with $m\ge N$ and $\ell\ge N$ .
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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-2026aAnalysisTopology
Let $(X,d)$ be a \reftext{def:metric-space-2026a}{metric space}, let $(x_m)_{m\in\mathbb{N}}$ be a \reftext{def:sequence-in-set-2026a}{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 every $m\in\mathbb{N}$ with $m\ge N$ . In this case we write $\lim_{m\to\infty}x_m=x$ and call $x$ the limit of $(x_m)$ in $(X,d)$ .
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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-2026aAnalysisSet 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-2026bAnalysisTopology
Let $(X,d)$ be a \reftext{def:complete-metric-space-2026a}{complete metric space}, and suppose that $X$ is nonempty. Let $T:X\to X$ be a \reftext{def:contraction-metric-space-2026a}{contraction}. Then $T$ has a unique \reftext{def:fixed-point-self-map-2026a}{fixed point} in $X$ . Moreover, for every $x_0\in X$ , the iterated \reftext{def:sequence-in-set-2026a}{sequence} $x_{m+1}=T(x_m)\qquad (m\in\mathbb{N}\cup\{0\})$ \reftext{def:convergent-sequence-metric-space-2026a}{converges} to that fixed point, where $\mathbb{N}$ denotes the \reftext{def:natural-numbers-2026a}{natural numbers}.
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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-2026aAnalysisTopology
Let $(X,d)$ be a \reftext{def:metric-space-2026a}{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-2026aAnalysisTopology
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-2026aAnalysisTopology
Let $(X,d)$ be a \reftext{def:metric-space-2026a}{metric space}. We say that $(X,d)$ is complete if every \reftext{def:cauchy-sequence-metric-space-2026a}{Cauchy sequence} in $(X,d)$ \reftext{def:convergent-sequence-metric-space-2026a}{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 \reftext{def:product-real-matrices-2026a}{the matrix product definition} and $I_n$ is the identity matrix from \ref{def:identity-matrix-2026a}. A real $n\times n$ matrix $A$ is called invertible if there exists an $n\times n$ real matrix $B$ that is an inverse of $A$ . If $A$ is invertible, its inverse is unique, and we denote the unique inverse by $A^{-1}.$
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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-2026aGeometryTopology
Let $M$ be a \reftext{def:smooth-manifold-with-boundary-2026a}{smooth manifold with boundary}. We say that $M$ is orientable if it admits an oriented smooth atlas in the sense of \ref{def:oriented-smooth-atlas-manifold-boundary-2026a}. An oriented smooth manifold with boundary is a smooth manifold with boundary together with a chosen oriented smooth atlas. Such a chosen oriented smooth atlas is called an orientation of $M$ .
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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-2026aGeometryTopology
Let $M$ be a \reftext{def:smooth-manifold-with-boundary-2026a}{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 positively compatible on the interior of $M$ in the sense of \ref{def:positive-compatibility-charts-interior-manifold-boundary-2026a}.
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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-2026aGeometryTopologyMultivariable Calculus
Let $M$ be a \reftext{def:smooth-manifold-with-boundary-2026a}{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 condition holds. For every point $x\in \varphi\bigl(U\cap V\cap \operatorname{int}(M)\bigr),$ consider any local smooth extension $F:W\to W'$ of the transition map $\psi\circ\varphi^{-1}$ at $x$ that is furnished by the smooth-compatibility condition from \ref{def:smooth-compatible-charts-upper-half-space-2026a}. Then $\det J_F(x)>0,$ where the Jacobian determinant is the one from \ref{def:jacobian-determinant-euclidean-open-set-2026a}. This condition is independent of the chosen local extension, because $x$ lies in the interior of the half-space chart image and any two such extensions agree on an open neighborhood of $x$ in $\mathbb{R}^n$ .
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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-2026aGeometryMultivariable Calculus
Let $n\in\mathbb{N}$ , let $U\subseteq\mathbb{R}^n$ be \reftext{def:open-subset-euclidean-space-2026a}{open}, let $F:U\to\mathbb{R}^n$ , and let $a\in U$ . Suppose that $F$ is differentiable at $a$ in the sense of \ref{def:differentiable-map-at-point-euclidean-2026a}, so that the Jacobian matrix $J_F(a)$ is defined. The determinant of this matrix, computed using \ref{def:determinant-real-square-matrix-2026a}, is denoted by $\det J_F(a)$ and is called the Jacobian determinant of $F$ at $a$ . If $F$ is \reftext{def:smooth-map-euclidean-open-set-2026a}{smooth} on $U$ , then the function $x\longmapsto \det J_F(x)$ from $U$ to $\mathbb{R}$ is called the Jacobian determinant function of $F$ .
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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 AlgebraMultivariable 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 \reftext{def:permutation-initial-segment-2026a}{the permutation definition} and $\operatorname{sgn}(\sigma)$ is the sign from \reftext{def:sign-permutation-2026a}{the sign definition}.
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T23:05:46.494304Z
Second Countable Topological Space
definitiondef:second-countable-topological-space-2026aTopology
Let $(X,\mathcal{T})$ be a \reftext{def:topological-space-2026a}{topological space}. We say that $X$ is second countable if there exists a basis $\mathcal{B}$ for $\mathcal{T}$ in the sense of \ref{def:basis-topology-2026a} such that $\mathcal{B}$ is countable.
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Authors ChatGPT-5.4, Aaron · Created 2026-04-16T22:26:34.298035Z

Showing 81-100 of 321

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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

Second Countable Topological Space

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

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