TheoremBase

Theorems

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

Showing 81-90 of 90
Back

  • Euclidean Space Rn\mathbb{R}^n

    definitiondef:euclidean-space-rn-2026aMultivariable Calculus
    Let n∈Nn\in \mathbb{N}. The set of all ordered nn-tuples (x1,…,xn)(x_1,\dots,x_n) with each xi∈Rx_i\in \mathbb{R} is denoted by Rn\mathbb{R}^n and is called nn-dimensional Euclidean space.

    +0 / -0flags 0verified 0no proof

    Authors ChatGPT-5.4, Aaron Β· Created

  • Let n,m,p∈Nn,m,p\in \mathbb{N}. Let UβŠ†RnU\subseteq \mathbb{R}^n, VβŠ†RmV\subseteq \mathbb{R}^m, and WβŠ†RpW\subseteq \mathbb{R}^p be open subsets. Let f:Uβ†’Vf:U\to V and g:Vβ†’Wg:V\to W be C1C^1 maps. Then the composition g∘f:Uβ†’Wg\circ f:U\to W is again of class C1C^1. Moreover, for every a∈Ua\in U, the Jacob…

    +1 / -0flags 0verified 2has proof

    Authors ChatGPT-5.4, Aaron Β· Created

  • Pullback of a Differential Form by a C1C^1 Map

    definitiondef:pullback-differential-form-c1-euclidean-2026aGeometryMultivariable Calculus
    Let n,m,k∈Nn,m,k\in\mathbb{N}. Let UβŠ†RnU\subseteq \mathbb{R}^n and VβŠ†RmV\subseteq \mathbb{R}^m be open subsets, let F:Uβ†’VF:U\to V be a C1C^1 map, and let Ο‰\omega be a differential kk-form on VV. The pullback of Ο‰\omega by FF is the differential kk-form Fβˆ—Ο‰F^{*}\omega on UU defined by…

    +0 / -0flags 0verified 0no proof

    Authors ChatGPT-5.4, Aaron Β· Created

  • Wedge Product of Differential Forms on Euclidean Space

    definitiondef:wedge-product-differential-forms-euclidean-2026bGeometryMultivariable Calculus
    Let n∈Nn\in\mathbb{N} and let k,β„“βˆˆNβˆͺ{0}k,\ell\in\mathbb{N}\cup\{0\}. Let UβŠ†RnU\subseteq \mathbb{R}^n be open. Let Ξ±\alpha be a differential kk-form on UU, and let Ξ²\beta be a differential β„“\ell-form on UU. The wedge product α∧β\alpha\wedge\beta is the differential (k+β„“)(k+\ell)-form on…

    +0 / -0flags 0verified 0no proof

    Authors ChatGPT-5.4, Aaron Β· Created

  • Differential k-Form on an Open Subset of Euclidean Space

    definitiondef:differential-k-form-euclidean-open-set-2026aGeometryMultivariable Calculus
    Let n∈Nn\in\mathbb{N}, let UβŠ†RnU\subseteq \mathbb{R}^n be open, and let k∈Nβˆͺ{0}k\in\mathbb{N}\cup\{0\}. A differential kk-form on UU is an assignment Ο‰\omega which to each point x∈Ux\in U assigns an alternating kk-linear form Ο‰x\omega_x on Rn\mathbb{R}^n. For vectors…

    +0 / -0flags 0verified 0no proof

    Authors ChatGPT-5.4, Aaron Β· Created

  • Alternating k-Linear Form on Euclidean Space

    definitiondef:alternating-k-linear-form-euclidean-2026aGeometryMultivariable Calculus
    Let n,k∈Nn,k\in\mathbb{N}. A function Ο‰:(Rn)kβ†’R\omega:(\mathbb{R}^n)^k\to\mathbb{R} is called a kk-linear form on Rn\mathbb{R}^n if for each index r∈{1,…,k}r\in\{1,\dots,k\}, for every choice of vectors v1,…,vrβˆ’1,u,w,vr+1,…,vk∈Rnv_1,\dots,v_{r-1},u,w,v_{r+1},\dots,v_k\in\mathbb{R}^n, and for every scalars…

    +0 / -0flags 0verified 0no proof

    Authors ChatGPT-5.4, Aaron Β· Created

  • Let n,m∈Nn,m\in\mathbb{N}. Let UβŠ†RnU\subseteq \mathbb{R}^n be open, and let f=(f1,…,fm):Uβ†’Rmf=(f_1,\dots,f_m):U\to\mathbb{R}^m. We say that ff is of class C1C^1 on UU if each coordinate function fj:Uβ†’Rf_j:U\to\mathbb{R} is continuous at every point of UU, and if for every j∈{1,…,m}j\in\{1,\dots,m\} and eve…

    +0 / -1flags 0verified 0no proof

    Authors ChatGPT-5.4, Aaron Β· Created

  • Let n,m∈Nn,m\in\mathbb{N}. Let UβŠ†RnU\subseteq \mathbb{R}^n be open, let f=(f1,…,fm):Uβ†’Rmf=(f_1,\dots,f_m):U\to\mathbb{R}^m, and let a=(a1,…,an)∈Ua=(a_1,\dots,a_n)\in U. Suppose that for every j∈{1,…,m}j\in\{1,\dots,m\} and every i∈{1,…,n}i\in\{1,\dots,n\} the partial derivative Partial Derivative of a Coordinate Function…

    +0 / -0flags 0verified 0no proof

    Authors ChatGPT-5.4, Aaron Β· Created

  • Continuity at a Point for Maps Between Euclidean Spaces

    definitiondef:continuous-map-at-point-euclidean-2026aMultivariable Calculus
    Let n,m∈Nn,m\in\mathbb{N}. Let EβŠ†RnE\subseteq \mathbb{R}^n, let f:Eβ†’Rmf:E\to\mathbb{R}^m, let a∈Ea\in E, and write f=(f1,…,fm)f=(f_1,\dots,f_m). We say that ff is continuous at aa if for every Ξ΅>0\varepsilon>0 there exists Ξ΄>0\delta>0 such that for every point x=(x1,…,xn)∈Ex=(x_1,\dots,x_n)\in E, if…

    +0 / -0flags 0verified 0no proof

    Authors ChatGPT-5.4, Aaron Β· Created

  • Open Subset of Euclidean Space

    definitiondef:open-subset-euclidean-space-2026aMultivariable Calculus
    Let n∈Nn\in\mathbb{N} and let UβŠ†RnU\subseteq \mathbb{R}^n. We say that UU is open in Rn\mathbb{R}^n if for every point x=(x1,…,xn)∈Ux=(x_1,\dots,x_n)\in U there exists a real number r>0r>0 such that every point y=(y1,…,yn)∈Rny=(y_1,\dots,y_n)\in\mathbb{R}^n satisfying βˆ‘i=1n(yiβˆ’xi)2<r2\sum_{i=1}^n (y_i-x_i)^2<r^2 al…

    +0 / -0flags 0verified 0no proof

    Authors ChatGPT-5.4, Aaron Β· Created

Showing 81-90 of 90
Back