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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. The empty set $\varnothing$ and the whole set $X$ belong to $\mathcal{T}$ . For every set $A$ and every \reftext{def:family-subfamily-subsets-set-2026a}{family of subsets of $X$ } $(U_a)_{a\in A}$ such that $U_a\in\mathcal{T}$ for every $a\in A$ , the union $\bigcup_{a\in A} U_a$ also belongs to $\mathcal{T}$ . For every \reftext{def:natural-numbers-2026a}{natural number} $n\in\mathbb{N}$ and every subsets $U_1,\dots,U_n\in\mathcal{T}$ , the intersection $\bigcap_{i=1}^n U_i$ also belongs to $\mathcal{T}$ . The elements of $\mathcal{T}$ are called open sets, and $\mathcal{T}$ is called a topology on $X$ .
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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 \ref{def:permutation-initial-segment-2026a}, and suppose that $\sigma=\tau_{r_1}\circ\cdots\circ\tau_{r_N},$ where each $\tau_{r_j}$ is an adjacent transposition as in \ref{thm:permutation-product-adjacent-transpositions-2026a}. Then $\operatorname{sgn}(\sigma)=(-1)^N,$ where $\operatorname{sgn}(\sigma)$ is the sign from \ref{def:sign-permutation-2026a}.
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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 \ref{def:permutation-initial-segment-2026a}. 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 $m\in\{1,\dots,n\}\setminus\{r,r+1\}$ . Then there exist an integer $N\ge 0$ and indices $r_1,\dots,r_N\in\{1,\dots,n-1\}$ such that $\sigma=\tau_{r_1}\circ\cdots\circ\tau_{r_N}.$
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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-2026aGeometryMultivariable Calculus
Let $n,k\in\mathbb{N}$ , let $U\subseteq \mathbb{R}^n$ be \reftext{def:open-subset-euclidean-space-2026a}{open}, let $1\le i_1<\cdots<i_k\le n$ , and let $\sigma\in S_k$ be a permutation in the sense of \ref{def:permutation-initial-segment-2026a}. Then the coordinate $1$ -forms from \ref{def:coordinate-1-form-euclidean-open-set-2026a} satisfy $dx_{i_{\sigma(1)}}\wedge\cdots\wedge dx_{i_{\sigma(k)}}=\operatorname{sgn}(\sigma)\,dx_{i_1}\wedge\cdots\wedge dx_{i_k},$ where $\operatorname{sgn}(\sigma)$ is the sign from \ref{def:sign-permutation-2026a} and the wedge products are taken in the sense of \ref{def:wedge-product-differential-forms-euclidean-2026b}.
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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-2026aGeometryMultivariable Calculus
Let $n,k\in\mathbb{N}$ with $1\le k\le n$ , let $U\subseteq \mathbb{R}^n$ be \reftext{def:open-subset-euclidean-space-2026a}{open}, let $\omega$ be a \reftext{def:c1-differential-k-form-euclidean-open-set-2026b}{ $C^1$ 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 \ref{thm:coordinate-expansion-differential-forms-euclidean-2026b} as $\omega=\sum_{1\le j_1<\cdots<j_{k-1}\le n} a_{j_1\dots j_{k-1}}\,dx_{j_1}\wedge\cdots\wedge dx_{j_{k-1}}.$ For each $r\in\{1,\dots,k\}$ , let $I^{\hat r}$ denote the increasing $(k-1)$ -tuple obtained from $(i_1,\dots,i_k)$ by omitting $i_r$ , and let $a_{I^{\hat r}}$ be the corresponding coefficient function. Then the coefficient of $dx_{i_1}\wedge\cdots\wedge dx_{i_k}$ in the coordinate expansion of $d\omega$ from \ref{def:exterior-derivative-c1-differential-form-euclidean-open-set-2026b} is $\sum_{r=1}^k (-1)^{r-1}\frac{\partial a_{I^{\hat r}}}{\partial x_{i_r}}.$
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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-2026aAnalysisMultivariable Calculus
Let $n\in\mathbb{N}$ , let $U\subseteq \mathbb{R}^n$ be \reftext{def:open-subset-euclidean-space-2026a}{open}, let $f:U\to\mathbb{R}$ be a \reftext{def:c1-map-euclidean-open-set-2026a}{ $C^1$ 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 $\{(x_1,\dots,x_{i-1},t,x_{i+1},\dots,x_n): t\in[a,b]\}\subseteq U.$ Define $g:[a,b]\to\mathbb{R}$ by $g(t)=f(x_1,\dots,x_{i-1},t,x_{i+1},\dots,x_n).$ Then $g$ is continuous on $[a,b]$ , differentiable at every $t\in(a,b)$ , and $g'(t)=\frac{\partial f}{\partial x_i}(x_1,\dots,x_{i-1},t,x_{i+1},\dots,x_n)$ for every $t\in(a,b)$ .
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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-2026bGeometryMultivariable Calculus
Let $n\in\mathbb{N}$ , let $U\subseteq \mathbb{R}^n$ be \reftext{def:open-subset-euclidean-space-2026a}{open}, let $k,\ell,m\in\mathbb{N}\cup\{0\}$ , let $\alpha$ be a \reftext{def:differential-k-form-euclidean-open-set-2026a}{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 $(\alpha\wedge\beta)\wedge\gamma=\alpha\wedge(\beta\wedge\gamma).$ Consequently, whenever $\omega_1,\dots,\omega_r$ are differential forms on $U$ for some $r\in\mathbb{N}$ , the expression $\omega_1\wedge\cdots\wedge\omega_r$ is unambiguous.
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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-2026bAnalysisGeometryMultivariable Calculus
Let $n,k\in\mathbb{N}$ with $1\le k\le n$ , let $(S,\varepsilon)$ be an \reftext{def:oriented-k-sub-rectangle-euclidean-2026a}{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 \reftext{def:c1-differential-k-form-euclidean-open-set-2026b}{ $C^1$ differential $(k-1)$ -form} on $U$ . Then $\int_{(S,\varepsilon)} d\omega = \sum_{(T,\eta)\in \partial(S,\varepsilon)} \int_{(T,\eta)} \omega,$ where $d\omega$ is the exterior derivative from \ref{def:exterior-derivative-c1-differential-form-euclidean-open-set-2026b}, the left-hand side and each term on the right are integrals in the sense of \ref{def:integral-form-oriented-k-sub-rectangle-euclidean-2026b}, and the boundary collection is the one from \ref{def:boundary-oriented-k-sub-rectangle-euclidean-2026a}.
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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-2026aGeometryMultivariable 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 $R=[a_1,b_1]\times\cdots\times[a_k,b_k]$ be the associated \reftext{def:standard-k-rectangle-euclidean-2026a}{standard $k$ -rectangle}. Define the coordinate insertion map $\lambda_R:R\to\mathbb{R}^n$ by sending $(t_1,\dots,t_k)$ to the point $x=(x_1,\dots,x_n)$ whose coordinates satisfy $x_{i_r}=t_r$ for $r\in\{1,\dots,k\}$ and $x_j=c_j$ for every remaining index $j$ . The image $S=\lambda_R(R)\subseteq \mathbb{R}^n$ is called a $k$ -sub-rectangle of $\mathbb{R}^n$ . An oriented $k$ -sub-rectangle of $\mathbb{R}^n$ is a pair $(S,\varepsilon)$ where $S$ is such a $k$ -sub-rectangle and $\varepsilon\in\{1,-1\}$ .
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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-2026bAnalysisGeometryMultivariable Calculus
Let $n,k\in\mathbb{N}$ with $k\le n$ , let $(S,\varepsilon)$ be an \reftext{def:oriented-k-sub-rectangle-euclidean-2026a}{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 \reftext{def:continuous-differential-k-form-euclidean-open-set-2026b}{continuous differential $k$ -form} on $U$ . Choose data as in \ref{def:oriented-k-sub-rectangle-euclidean-2026a}, so that $S=\lambda_R(R)$ for a standard $k$ -rectangle $R=[a_1,b_1]\times\cdots\times[a_k,b_k]$ and active coordinate indices $1\le i_1<\cdots<i_k\le n$ . Write $\omega$ in the coordinate expansion from \ref{thm:coordinate-expansion-differential-forms-euclidean-2026b}. Let $f:U\to\mathbb{R}$ be the coefficient function of the basis form $dx_{i_1}\wedge\cdots\wedge dx_{i_k}.$ For each fixed $(t_2,\dots,t_k)\in [a_2,b_2]\times\cdots\times[a_k,b_k]$ , define $F_1(t_2,\dots,t_k)=\int_{a_1}^{b_1} f(\lambda_R(t_1,t_2,\dots,t_k))\,dt_1,$ where this is the one-dimensional \reftext{def:riemann-integrable-closed-interval-c54-2026b}{Riemann integral} on $[a_1,b_1]$ ; it exists because the integrand is continuous on $[a_1,b_1]$ , hence \reftext{lem:continuous-implies-riemann-integrable-c54-2026b}{Riemann integrable}. Recursively, for each $r\in\{2,\dots,k\}$ and each fixed $(t_{r+1},\dots,t_k)\in [a_{r+1},b_{r+1}]\times\cdots\times[a_k,b_k]$ , define $F_r(t_{r+1},\dots,t_k)=\int_{a_r}^{b_r} F_{r-1}(t_r,\dots,t_k)\,dt_r,$ again in the one-dimensional Riemann sense of \ref{def:riemann-integrable-closed-interval-c54-2026b}, whenever the integrand is viewed as a function of $t_r$ alone. The integral of $\omega$ over $(S,\varepsilon)$ is defined by $\int_{(S,\varepsilon)} \omega=\varepsilon F_k.$ Equivalently, $\int_{(S,\varepsilon)} \omega = \varepsilon \int_{a_k}^{b_k}\cdots\int_{a_1}^{b_1} f(\lambda_R(t_1,\dots,t_k))\, dt_1\cdots dt_k,$ with the right-hand side understood through the recursive construction above.
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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-2026aGeometryMultivariable Calculus
Let $n,k\in\mathbb{N}$ with $1\le k\le n$ , and let $(S,\varepsilon)$ be an \reftext{def:oriented-k-sub-rectangle-euclidean-2026a}{oriented $k$ -sub-rectangle} of $\mathbb{R}^n$ . Choose data as in \ref{def:oriented-k-sub-rectangle-euclidean-2026a}, so that $S=\lambda_R(R)$ for a standard $k$ -rectangle $R=[a_1,b_1]\times\cdots\times[a_k,b_k].$ For each index $r\in\{1,\dots,k\}$ , let $R_r=[a_1,b_1]\times\cdots\times[a_{r-1},b_{r-1}]\times[a_{r+1},b_{r+1}]\times\cdots\times[a_k,b_k],$ and define insertion maps $\mu_r^-:R_r\to R$ and $\mu_r^+:R_r\to R$ by inserting $a_r$ or $b_r$ in the $r$ th coordinate. Composing with $\lambda_R$ gives $(k-1)$ -sub-rectangles $S_r^-=(\lambda_R\circ\mu_r^-)(R_r),\qquad S_r^+=(\lambda_R\circ\mu_r^+)(R_r).$ The boundary of $(S,\varepsilon)$ is the collection of oriented $(k-1)$ -sub-rectangles $\partial(S,\varepsilon)=\{(S_r^+,\varepsilon(-1)^{r-1}),(S_r^-,-\varepsilon(-1)^{r-1}): r\in\{1,\dots,k\}\}.$
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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-2026aGeometryMultivariable 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-2026bGeometryMultivariable Calculus
Let $n\in\mathbb{N}$ , let $U\subseteq \mathbb{R}^n$ be \reftext{def:open-subset-euclidean-space-2026a}{open}, let $k\in\mathbb{N}\cup\{0\}$ , and let $\omega$ be a \reftext{def:c1-differential-k-form-euclidean-open-set-2026b}{ $C^1$ 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 \ref{thm:coordinate-expansion-differential-forms-euclidean-2026b}. The exterior derivative of $\omega$ is the differential $(k+1)$ -form $d\omega$ on $U$ defined by $d\omega=\sum_{1\le i_1<\cdots<i_k\le n}\;\sum_{j=1}^n \frac{\partial a_{i_1\dots i_k}}{\partial x_j}\, dx_j\wedge dx_{i_1}\wedge\cdots\wedge dx_{i_k},$ where the partial derivatives are in the sense of \ref{def:partial-derivative-coordinate-map-2026a}. When $k=0$ , this is the usual differential of a $C^1$ real-valued function.
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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-2026bGeometryMultivariable Calculus
Let $n\in\mathbb{N}$ , let $U\subseteq \mathbb{R}^n$ be \reftext{def:open-subset-euclidean-space-2026a}{open}, let $k\in\mathbb{N}\cup\{0\}$ , and let $\omega$ be a \reftext{def:differential-k-form-euclidean-open-set-2026a}{differential $k$ -form} on $U$ . Write $\omega$ in the coordinate expansion from \ref{thm:coordinate-expansion-differential-forms-euclidean-2026b}. We say that $\omega$ is of class $C^1$ if each coefficient function in that expansion is a \reftext{def:c1-map-euclidean-open-set-2026a}{ $C^1$ map} from $U$ to $\mathbb{R}$ .
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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-2026bGeometryMultivariable Calculus
Let $n\in\mathbb{N}$ , let $U\subseteq \mathbb{R}^n$ be \reftext{def:open-subset-euclidean-space-2026a}{open}, let $k\in\mathbb{N}\cup\{0\}$ , and let $\omega$ be a \reftext{def:differential-k-form-euclidean-open-set-2026a}{differential $k$ -form} on $U$ . Write $\omega$ in the coordinate expansion from \ref{thm:coordinate-expansion-differential-forms-euclidean-2026b}. We say that $\omega$ is continuous if each coefficient function in that expansion is \reftext{def:continuous-map-at-point-euclidean-2026a}{continuous at every point of $U$ }.
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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-2026bGeometryMultivariable Calculus
Let $n\in\mathbb{N}$ , let $U\subseteq \mathbb{R}^n$ be \reftext{def:open-subset-euclidean-space-2026a}{open}, let $k\in\mathbb{N}\cup\{0\}$ , and let $\omega$ be a \reftext{def:differential-k-form-euclidean-open-set-2026a}{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 $(i_1,\dots,i_k)$ with $1\le i_1<\cdots<i_k\le n$ such that $\omega=\sum_{1\le i_1<\cdots<i_k\le n} a_{i_1\dots i_k}\, dx_{i_1}\wedge\cdots\wedge dx_{i_k},$ where each $dx_i$ is the coordinate $1$ -form from \ref{def:coordinate-1-form-euclidean-open-set-2026a} and the wedge product is the one from \ref{def:wedge-product-differential-forms-euclidean-2026b}. When $k=0$ , this says that $\omega$ is uniquely equal to a real-valued function on $U$ .
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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-2026aGeometryMultivariable Calculus
Let $n\in\mathbb{N}$ , let $U\subseteq \mathbb{R}^n$ be \reftext{def:open-subset-euclidean-space-2026a}{open}, and let $i\in\{1,\dots,n\}$ . The $i$ th coordinate $1$ -form on $U$ is the \reftext{def:differential-k-form-euclidean-open-set-2026a}{differential $1$ -form} $dx_i$ on $U$ defined by $(dx_i)_x(v)=v_i$ for every point $x\in U$ and every vector $v=(v_1,\dots,v_n)\in\mathbb{R}^n$ .
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Authors ChatGPT-5.4, Aaron · Created 2026-04-08T21:13:28.050001Z
$C^1$ Maps on Euclidean Open Sets are Differentiable
theoremthm:c1-implies-differentiable-euclidean-2026aMultivariable Calculus
Let $n,m\in\mathbb{N}$ . Let $U\subseteq \mathbb{R}^n$ be \reftext{def:open-subset-euclidean-space-2026a}{open}, and let $f:U\to\mathbb{R}^m$ be a \reftext{def:c1-map-euclidean-open-set-2026a}{ $C^1$ map}. Then for every point $a\in U$ , the map $f$ is \reftext{def:differentiable-map-at-point-euclidean-2026a}{differentiable} at $a$ .
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T21:14:19.973687Z
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 \reftext{def:continuous-map-at-point-euclidean-2026a}{continuous} at $a$ and that $g$ is continuous at $f(a)$ . Then the composition $g\circ f:E\to\mathbb{R}^p$ is continuous at $a$ .
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T21:14:19.915854Z
Sums and Products of Continuous Real-Valued Functions
theoremthm:sum-product-continuous-real-2026aAnalysis
Let $E\subseteq \mathbb{R}^n$ , let $u,v:E\to\mathbb{R}$ , and let $a\in E$ . Suppose that $u$ and $v$ are \reftext{def:continuous-at-point-c54-2026b}{continuous} at $a$ . Then the functions $u+v:E\to\mathbb{R}$ and $uv:E\to\mathbb{R}$ are continuous at $a$ .
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Authors ChatGPT-5.4, Aaron · Created 2026-04-06T21:14:19.834706Z

Showing 141-160 of 321

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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

C1C^1C1 Maps on Euclidean Open Sets are Differentiable

Composition of Continuous Euclidean Maps

Sums and Products of Continuous Real-Valued Functions

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

$C^1$ Maps on Euclidean Open Sets are Differentiable