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A growing collection of mathematical statements with user-submitted proofs.

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  • Independence of the Manifold Integral from Chart and Partition Choices

    theoremthm:integral-manifold-independence-choices-2026aAnalysisGeometryMultivariable Calculus
    Let n∈n\in \reftext{def:natural-numbers-2026a}{N\mathbb{N}} and let MM be an \reftext{def:oriented-smooth-manifold-boundary-2026a}{oriented smooth manifold with boundary} of dimension nn that is compact in the sense of \ref{def:smooth-manifold-with-boundary-2026a}, with chosen \reftext{def:oriented-smooth-atlas-manifold-boundary-2026a}{oriented smooth atlas} ((Uα,φα))α∈A((U_\alpha,\varphi_\alpha))_{\alpha\in A}, and write Ωα=φα(Uα)\Omega_\alpha=\varphi_\alpha(U_\alpha). Let ω=(ωα)α∈A\omega=(\omega_\alpha)_{\alpha\in A} be a \reftext{def:smooth-differential-k-form-manifold-boundary-2026a}{smooth differential nn-form} on MM. Let N∈NN\in\mathbb{N}, α1,…,αN∈A\alpha_1,\dots,\alpha_N\in A, and let χ1,…,χN\chi_1,\dots,\chi_N be a smooth partition of unity subordinate to Uα1,…,UαNU_{\alpha_1},\dots,U_{\alpha_N} as in \ref{thm:smooth-partition-unity-compact-manifold-boundary-2026a}. For each i∈{1,…,N}i\in\{1,\dots,N\}, let χiω\chi_i\omega denote the \reftext{def:smooth-differential-k-form-manifold-boundary-2026a}{smooth differential nn-form} on MM whose chart representative in each chart (Uα,φα)(U_\alpha,\varphi_\alpha) assigns to x∈Ωαx\in\Omega_\alpha the pointwise scalar multiple of the alternating form ωα,x\omega_{\alpha,x} by the real number χi,α(x)\chi_{i,\alpha}(x), where χi,α\chi_{i,\alpha} is the chart representative of χi\chi_i. Then the following hold. For each i∈{1,…,N}i\in\{1,\dots,N\}, the representative (χiω)αi(\chi_i\omega)_{\alpha_i} of χiω\chi_i\omega in the chart (Uαi,φαi)(U_{\alpha_i},\varphi_{\alpha_i}) is a continuous differential nn-form on the admissible domain Ωαi\Omega_{\alpha_i} that is compactly supported in Ωαi\Omega_{\alpha_i}, in the sense of \ref{def:continuous-n-form-support-euclidean-domain-2026a}. The real number S=∑i=1N∫Ωαi(χiω)αi,S=\sum_{i=1}^{N}\int_{\Omega_{\alpha_i}}(\chi_i\omega)_{\alpha_i}, with each summand an integral in the sense of \ref{def:integral-compactly-supported-n-form-euclidean-2026a}, is well defined. The value SS does not depend on the choices made: if N′∈NN'\in\mathbb{N}, β1,…,βN′∈A\beta_1,\dots,\beta_{N'}\in A, and χ1′,…,χN′′\chi_1',\dots,\chi_{N'}' form another smooth partition of unity subordinate to Uβ1,…,UβN′U_{\beta_1},\dots,U_{\beta_{N'}} as in \ref{thm:smooth-partition-unity-compact-manifold-boundary-2026a}, then ∑i=1N∫Ωαi(χiω)αi=∑j=1N′∫Ωβj(χj′ω)βj.\sum_{i=1}^{N}\int_{\Omega_{\alpha_i}}(\chi_i\omega)_{\alpha_i}=\sum_{j=1}^{N'}\int_{\Omega_{\beta_j}}(\chi_j'\omega)_{\beta_j}. The proof of claim 3 rests on the pullback invariance of the integral under orientation-preserving smooth diffeomorphisms, \ref{thm:pullback-invariance-integral-diffeomorphism-euclidean-2026a}, applied to the transition maps of the oriented atlas.

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  • Smooth Partitions of Unity on a Compact Smooth Manifold with Boundary

    theoremthm:smooth-partition-unity-compact-manifold-boundary-2026aAnalysisGeometryTopology
    Let MM be a \reftext{def:smooth-manifold-with-boundary-2026a}{smooth manifold with boundary} that is compact in the sense of that definition, with chosen smooth atlas ((Uα,φα))α∈A((U_\alpha,\varphi_\alpha))_{\alpha\in A}. Then there exist N∈N\in \reftext{def:natural-numbers-2026a}{N\mathbb{N}}, indices α1,…,αN∈A\alpha_1,\dots,\alpha_N\in A, and \reftext{def:smooth-differential-k-form-manifold-boundary-2026a}{smooth differential 00-forms} χ1,…,χN\chi_1,\dots,\chi_N on MM, identified with real-valued functions on MM as in that definition, such that the following hold. 0≤χi(p)≤10\le\chi_i(p)\le 1 for every i∈{1,…,N}i\in\{1,\dots,N\} and every p∈Mp\in M. For each i∈{1,…,N}i\in\{1,\dots,N\}, the \textbf{support} of χi\chi_i — the set of all p∈Mp\in M such that every open subset of MM containing pp contains a point qq with χi(q)≠0\chi_i(q)\ne 0 — is contained in UαiU_{\alpha_i}. For every p∈Mp\in M, ∑i=1Nχi(p)=1.\sum_{i=1}^{N}\chi_i(p)=1. A family χ1,…,χN\chi_1,\dots,\chi_N with these properties is called a \textbf{smooth partition of unity subordinate to the chart domains} Uα1,…,UαNU_{\alpha_1},\dots,U_{\alpha_N}.

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  • Restriction of a Smooth Differential Form to the Boundary

    definitiondef:restriction-form-boundary-manifold-2026aGeometryTopologyMultivariable Calculus
    Let n∈n\in \reftext{def:natural-numbers-2026a}{N\mathbb{N}} with n≥2n\ge 2, let MM be a \reftext{def:smooth-manifold-with-boundary-2026a}{smooth manifold with boundary} of dimension nn with nonempty \reftext{def:boundary-smooth-manifold-with-boundary-2026a}{boundary} ∂M\partial M, and equip ∂M\partial M with the smooth manifold structure of dimension n−1n-1 from \ref{thm:boundary-smooth-manifold-structure-2026a} (if MM is oriented, one may equally use the atlas of the \reftext{def:induced-orientation-boundary-manifold-2026a}{induced orientation}, whose chart maps differ from induced boundary charts at most by composition with the reflection ρ\rho appearing there). Let k∈N∪{0}k\in\mathbb{N}\cup\{0\} and let ω=(ωα)α∈A\omega=(\omega_\alpha)_{\alpha\in A} be a \reftext{def:smooth-differential-k-form-manifold-boundary-2026a}{smooth differential kk-form} on MM, relative to the chosen atlas ((Uα,φα))α∈A((U_\alpha,\varphi_\alpha))_{\alpha\in A} of MM. Let (U′∩∂M,ψ)(U'\cap\partial M,\psi) be a chart of the chosen atlas of ∂M\partial M, induced as in \ref{thm:boundary-smooth-manifold-structure-2026a} from a chart (Uα,φα)(U_\alpha,\varphi_\alpha) of MM (possibly composed with the reflection ρ\rho), and let Ωψ\Omega_\psi denote its image. Between the \reftext{def:euclidean-space-rn-2026a}{Euclidean spaces} Rn−1\mathbb{R}^{n-1} and Rn\mathbb{R}^n, let j:Rn−1→Rnj:\mathbb{R}^{n-1}\to\mathbb{R}^n be the unique affine map satisfying j(y)=φα(ψ−1(y))j(y)=\varphi_\alpha(\psi^{-1}(y)) for every y∈Ωψy\in\Omega_\psi; it is the composition of the inverse translation (and, where applicable, the reflection ρ\rho) with the inclusion of Rn−1\mathbb{R}^{n-1} into the boundary hyperplane of the \reftext{def:closed-upper-half-space-euclidean-2026a}{closed upper half-space} HnH^n. Since jj is affine, it is a \reftext{def:smooth-map-euclidean-open-set-2026a}{smooth map} and its \reftext{def:differentiable-map-at-point-euclidean-2026a}{Jacobian matrix} JjJ_j is a constant n×(n−1)n\times(n-1) matrix. The \textbf{restriction of ω\omega to ∂M\partial M}, denoted ι∗ω\iota^{*}\omega where ι:∂M→M\iota:\partial M\to M is the inclusion map, is the family whose representative in the chart (U′∩∂M,ψ)(U'\cap\partial M,\psi) assigns to each y∈Ωψy\in\Omega_\psi the \reftext{def:alternating-k-linear-form-euclidean-2026a}{alternating kk-linear form} on Rn−1\mathbb{R}^{n-1} given by (ι∗ω)ψ,y(v1,…,vk)=ωα, j(y)(Jjv1,…,Jjvk)(\iota^{*}\omega)_{\psi,y}(v_1,\dots,v_k)=\omega_{\alpha,\,j(y)}\bigl(J_j v_1,\dots,J_j v_k\bigr) for all vectors v1,…,vk∈Rn−1v_1,\dots,v_k\in\mathbb{R}^{n-1}, where JjvrJ_j v_r is the \reftext{def:matrix-vector-product-2026a}{matrix-vector product}; this is the pullback formula of \ref{def:pullback-differential-form-c1-euclidean-2026a} applied to the affine map jj. The family of these representatives is a \reftext{def:smooth-differential-k-form-manifold-boundary-2026a}{smooth differential kk-form} on ∂M\partial M.

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  • Induced Orientation on the Boundary of an Oriented Smooth Manifold with Boundary

    definitiondef:induced-orientation-boundary-manifold-2026aGeometryTopologyMultivariable Calculus
    Let n∈n\in \reftext{def:natural-numbers-2026a}{N\mathbb{N}} with n≥2n\ge 2, and let MM be an \reftext{def:oriented-smooth-manifold-boundary-2026a}{oriented smooth manifold with boundary} of dimension nn with nonempty \reftext{def:boundary-smooth-manifold-with-boundary-2026a}{boundary} ∂M\partial M. Equip ∂M\partial M with the smooth manifold structure of dimension n−1n-1 from \ref{thm:boundary-smooth-manifold-structure-2026a}, built from the induced boundary charts arising from charts of the chosen oriented atlas. Let ρ:Rn−1→Rn−1\rho:\mathbb{R}^{n-1}\to\mathbb{R}^{n-1} denote the reflection of \reftext{def:euclidean-space-rn-2026a}{Euclidean space} given by ρ(y1,y2,…,yn−1)=(−y1,y2,…,yn−1).\rho(y_1,y_2,\dots,y_{n-1})=(-y_1,y_2,\dots,y_{n-1}). When nn is odd we have n−1≥2n-1\ge 2, so ρ\rho leaves the last coordinate unchanged and maps the region of points of Hn−1H^{n-1} with positive last coordinate into itself; hence composing a chart map with ρ\rho again yields a \reftext{def:chart-upper-half-space-2026a}{chart}. The \textbf{induced orientation} on ∂M\partial M is the orientation of ∂M\partial M, in the sense of \ref{def:oriented-smooth-manifold-boundary-2026a}, given by the following \reftext{def:oriented-smooth-atlas-manifold-boundary-2026a}{oriented smooth atlas}, where \reftext{def:even-odd-natural-numbers-2026a}{even and odd} refer to the parity of nn. If nn is even: the atlas of all induced boundary charts (U′∩∂M,ψ)(U'\cap\partial M,\psi) from \ref{thm:boundary-smooth-manifold-structure-2026a} arising from charts of the chosen oriented atlas of MM. This is an oriented smooth atlas by \ref{lem:positive-jacobian-boundary-transition-2026a}. If nn is odd: the atlas of all charts (U′∩∂M,ρ∘ψ)(U'\cap\partial M,\rho\circ\psi), where (U′∩∂M,ψ)(U'\cap\partial M,\psi) ranges over the induced boundary charts as in case 1. This is an oriented smooth atlas because each transition map of this atlas is the conjugate by ρ\rho of the corresponding transition map of the atlas in case 1, and conjugation by ρ\rho does not change the Jacobian determinant of the transition, which is positive by \ref{lem:positive-jacobian-boundary-transition-2026a}. This sign convention is chosen so that the Stokes identity holds without an extraneous sign; it corresponds to the outward-normal-first convention for orienting the boundary.

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  • Transition Maps of Induced Boundary Charts of an Oriented Atlas are Orientation-Preserving

    lemmalem:positive-jacobian-boundary-transition-2026aGeometryTopologyMultivariable Calculus
    Let n∈n\in \reftext{def:natural-numbers-2026a}{N\mathbb{N}} with n≥2n\ge 2, and let MM be an \reftext{def:oriented-smooth-manifold-boundary-2026a}{oriented smooth manifold with boundary} of dimension nn with nonempty \reftext{def:boundary-smooth-manifold-with-boundary-2026a}{boundary} ∂M\partial M, the orientation being given by the chosen \reftext{def:oriented-smooth-atlas-manifold-boundary-2026a}{oriented smooth atlas}. Then the following hold. Let (U,φ)(U,\varphi) and (V,ψ)(V,\psi) be charts of the oriented atlas with U∩V≠∅U\cap V\ne\varnothing, let a∈φ(U∩V)a\in\varphi(U\cap V), and let FF be any local smooth extension of the transition map ψ∘φ−1\psi\circ\varphi^{-1} at aa furnished by \ref{def:smooth-compatible-charts-upper-half-space-2026a}. Then the \reftext{def:jacobian-determinant-euclidean-open-set-2026a}{Jacobian determinant} satisfies det⁡JF(a)>0\det J_F(a)>0. (For aa in the image of the interior of MM this is the \reftext{def:positive-compatibility-charts-interior-manifold-boundary-2026a}{positive compatibility} required of the oriented atlas; the content of this claim is that positivity extends to all points of φ(U∩V)\varphi(U\cap V), including boundary points.) In the setting of claim 1, suppose additionally that aa lies in the boundary hyperplane of the \reftext{def:closed-upper-half-space-euclidean-2026a}{closed upper half-space} HnH^n, and write F=(F1,…,Fn)F=(F_1,\dots,F_n) in coordinates. Then the \reftext{def:partial-derivative-coordinate-map-2026a}{partial derivatives} of the last coordinate function satisfy ∂Fn∂xj(a)=0for j∈{1,…,n−1},∂Fn∂xn(a)>0.\frac{\partial F_n}{\partial x_j}(a)=0 \quad\text{for } j\in\{1,\dots,n-1\},\qquad \frac{\partial F_n}{\partial x_n}(a)>0. Let ψ1\psi_1 and ψ2\psi_2 be induced boundary charts of ∂M\partial M in the sense of \ref{thm:boundary-smooth-manifold-structure-2026a}, arising from charts of the oriented atlas, with overlapping domains. Then the transition map ψ2∘ψ1−1\psi_2\circ\psi_1^{-1} between the corresponding images is a \reftext{def:smooth-diffeomorphism-euclidean-half-space-domain-2026a}{smooth diffeomorphism} of admissible domains in \reftext{def:euclidean-space-rn-2026a}{Euclidean space} Rn−1\mathbb{R}^{n-1}, and it is orientation-preserving in the sense of that definition; that is, the determinant of its Jacobian matrix is positive at every point of its domain.

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  • Boundary of a Smooth Manifold with Boundary as a Smooth Manifold of Dimension n-1

    theoremthm:boundary-smooth-manifold-structure-2026aGeometryTopologyMultivariable Calculus
    Let n∈n\in \reftext{def:natural-numbers-2026a}{N\mathbb{N}} with n≥2n\ge 2, and let MM be a \reftext{def:smooth-manifold-with-boundary-2026a}{smooth manifold with boundary} of dimension nn whose \reftext{def:boundary-smooth-manifold-with-boundary-2026a}{boundary} ∂M\partial M is nonempty. Give ∂M\partial M the \reftext{def:subspace-topology-2026a}{subspace topology} inherited from MM. Then the following hold. ∂M\partial M is a \reftext{def:closed-subset-topological-space-2026a}{closed subset} of MM. With the subspace topology, ∂M\partial M is \reftext{def:hausdorff-topological-space-2026a}{Hausdorff} and \reftext{def:second-countable-topological-space-2026a}{second countable}. Let (U,φ)(U,\varphi) be a chart in the chosen atlas of MM with U∩∂M≠∅U\cap\partial M\ne\varnothing. Then φ(U∩∂M)=φ(U)∩∂Hn\varphi(U\cap\partial M)=\varphi(U)\cap\partial H^n, where ∂Hn\partial H^n is the boundary hyperplane of the \reftext{def:closed-upper-half-space-euclidean-2026a}{closed upper half-space}. Let π:Rn→Rn−1\pi:\mathbb{R}^n\to\mathbb{R}^{n-1}, between \reftext{def:euclidean-space-rn-2026a}{Euclidean spaces}, denote the projection π(x1,…,xn)=(x1,…,xn−1)\pi(x_1,\dots,x_n)=(x_1,\dots,x_{n-1}), and for c∈Rn−1c\in\mathbb{R}^{n-1} let Tc:Rn−1→Rn−1T_c:\mathbb{R}^{n-1}\to\mathbb{R}^{n-1} denote the translation Tc(y)=y+cT_c(y)=y+c. Then for every p∈U∩∂Mp\in U\cap\partial M there exist an open subset U′U' of MM with p∈U′⊆Up\in U'\subseteq U and a vector c∈Rn−1c\in\mathbb{R}^{n-1} such that the pair (U′∩∂M,ψ)(U'\cap\partial M,\psi), with chart map ψ=Tc∘π∘φ\psi=T_c\circ\pi\circ\varphi restricted to U′∩∂MU'\cap\partial M, is a \reftext{def:chart-upper-half-space-2026a}{chart} of dimension n−1n-1 on ∂M\partial M whose image is contained in the set of points y=(y1,…,yn−1)y=(y_1,\dots,y_{n-1}) of the closed upper half-space Hn−1H^{n-1} with yn−1>0y_{n-1}>0. Charts of this form are called \textbf{induced boundary charts} of ∂M\partial M. Any two induced boundary charts are \reftext{def:smooth-compatible-charts-upper-half-space-2026a}{smoothly compatible}, and the collection of all induced boundary charts is a \reftext{def:smooth-manifold-with-boundary-2026a}{smooth atlas} of dimension n−1n-1 on ∂M\partial M, making ∂M\partial M a smooth manifold with boundary of dimension n−1n-1 in which every point is an interior point in the sense of \ref{def:boundary-smooth-manifold-with-boundary-2026a}; that is, the boundary of ∂M\partial M is empty. If MM is compact in the sense of \ref{def:smooth-manifold-with-boundary-2026a}, then ∂M\partial M is compact, by claim 1 together with \ref{thm:closed-subset-compact-is-compact-2026a}.

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  • Exterior Derivative of a Smooth Differential Form on a Smooth Manifold with Boundary

    definitiondef:exterior-derivative-smooth-form-manifold-boundary-2026aGeometryTopologyMultivariable Calculus
    Let n∈n\in \reftext{def:natural-numbers-2026a}{N\mathbb{N}}, let MM be a \reftext{def:smooth-manifold-with-boundary-2026a}{smooth manifold with boundary} of dimension nn with chosen smooth atlas ((Uα,φα))α∈A((U_\alpha,\varphi_\alpha))_{\alpha\in A}, write Ωα=φα(Uα)\Omega_\alpha=\varphi_\alpha(U_\alpha), let k∈N∪{0}k\in\mathbb{N}\cup\{0\}, and let ω=(ωα)α∈A\omega=(\omega_\alpha)_{\alpha\in A} be a \reftext{def:smooth-differential-k-form-manifold-boundary-2026a}{smooth differential kk-form} on MM. Fix α∈A\alpha\in A and x∈Ωαx\in\Omega_\alpha. Choose an \reftext{def:open-subset-euclidean-space-2026a}{open} set WW in \reftext{def:euclidean-space-rn-2026a}{Euclidean space} Rn\mathbb{R}^n with x∈Wx\in W and a differential kk-form η\eta on WW with smooth coefficients agreeing with ωα\omega_\alpha on W∩ΩαW\cap\Omega_\alpha, as in the local smoothness property of \ref{def:smooth-differential-k-form-manifold-boundary-2026a}. Since smooth coefficient functions are in particular \reftext{def:c1-map-euclidean-open-set-2026a}{C1C^1 maps}, η\eta is a \reftext{def:c1-differential-k-form-euclidean-open-set-2026b}{C1C^1 differential kk-form} on WW, so its \reftext{def:exterior-derivative-c1-differential-form-euclidean-open-set-2026b}{exterior derivative} dηd\eta is defined. Set (dω)α,x=(dη)x.(d\omega)_{\alpha,x}=(d\eta)_x. This value does not depend on the choice of WW and η\eta: the coefficient functions of any two such local forms agree on a neighborhood of xx that is open in the \reftext{def:closed-upper-half-space-euclidean-2026a}{closed upper half-space} HnH^n, and the first-order partial derivatives of smooth functions at a point of such a set are determined by continuity from the values of the functions on that set. The family ((dω)α)α∈A((d\omega)_\alpha)_{\alpha\in A} so defined is a \reftext{def:smooth-differential-k-form-manifold-boundary-2026a}{smooth differential (k+1)(k+1)-form} on MM; its compatibility property holds because pullback commutes with the exterior derivative, by \ref{thm:pullback-commutes-exterior-derivative-euclidean-2026a}. This smooth (k+1)(k+1)-form is called the \textbf{exterior derivative} of ω\omega and is denoted dωd\omega.

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  • Smooth Differential k-Form on a Smooth Manifold with Boundary

    definitiondef:smooth-differential-k-form-manifold-boundary-2026aGeometryTopologyMultivariable Calculus
    Let n∈n\in \reftext{def:natural-numbers-2026a}{N\mathbb{N}}, and let MM be a \reftext{def:smooth-manifold-with-boundary-2026a}{smooth manifold with boundary} of dimension nn, with chosen smooth atlas ((Uα,φα))α∈A((U_\alpha,\varphi_\alpha))_{\alpha\in A}; for each α∈A\alpha\in A write Ωα=φα(Uα)\Omega_\alpha=\varphi_\alpha(U_\alpha). Each Ωα\Omega_\alpha is open in the \reftext{def:closed-upper-half-space-euclidean-2026a}{closed upper half-space} HnH^n of \reftext{def:euclidean-space-rn-2026a}{Euclidean space} Rn\mathbb{R}^n, hence an admissible domain in the sense of \ref{def:continuous-n-form-support-euclidean-domain-2026a}. Let k∈N∪{0}k\in\mathbb{N}\cup\{0\}. A \textbf{smooth differential kk-form} ω\omega on MM is a family (ωα)α∈A(\omega_\alpha)_{\alpha\in A} with the following three properties. For each α∈A\alpha\in A, ωα\omega_\alpha assigns to each point x∈Ωαx\in\Omega_\alpha an \reftext{def:alternating-k-linear-form-euclidean-2026a}{alternating kk-linear form} ωα,x\omega_{\alpha,x} on Rn\mathbb{R}^n. The assignment ωα\omega_\alpha is called the \textbf{chart representative} of ω\omega in the chart (Uα,φα)(U_\alpha,\varphi_\alpha). (Local smoothness.) For each α∈A\alpha\in A and each a∈Ωαa\in\Omega_\alpha there exist an \reftext{def:open-subset-euclidean-space-2026a}{open} set W⊆RnW\subseteq\mathbb{R}^n with a∈Wa\in W and a \reftext{def:differential-k-form-euclidean-open-set-2026a}{differential kk-form} η\eta on WW whose coefficient functions in the \reftext{thm:coordinate-expansion-differential-forms-euclidean-2026b}{coordinate expansion} are \reftext{def:smooth-map-euclidean-open-set-2026a}{smooth}, such that ηx=ωα,x\eta_x=\omega_{\alpha,x} for every x∈W∩Ωαx\in W\cap\Omega_\alpha. (Compatibility.) For all α,β∈A\alpha,\beta\in A with Uα∩Uβ≠∅U_\alpha\cap U_\beta\ne\varnothing, consider the transition map τβα=φβ∘φα−1\tau_{\beta\alpha}=\varphi_\beta\circ\varphi_\alpha^{-1} from φα(Uα∩Uβ)\varphi_\alpha(U_\alpha\cap U_\beta) to φβ(Uα∩Uβ)\varphi_\beta(U_\alpha\cap U_\beta); it is a \reftext{def:smooth-diffeomorphism-euclidean-half-space-domain-2026a}{smooth diffeomorphism} of admissible domains, its local smooth extensions being furnished by the \reftext{def:smooth-compatible-charts-upper-half-space-2026a}{smooth compatibility} of the charts. The requirement is that the \reftext{def:smooth-diffeomorphism-euclidean-half-space-domain-2026a}{pullback} satisfies τβα∗(ωβ)=ωα\tau_{\beta\alpha}^{*}(\omega_\beta)=\omega_\alpha on φα(Uα∩Uβ)\varphi_\alpha(U_\alpha\cap U_\beta), where ωβ\omega_\beta is restricted to φβ(Uα∩Uβ)\varphi_\beta(U_\alpha\cap U_\beta). When k=0k=0, each ωα\omega_\alpha is a real-valued function on Ωα\Omega_\alpha and property 3 states that ωα=ωβ∘τβα\omega_\alpha=\omega_\beta\circ\tau_{\beta\alpha} on chart overlaps. Consequently there is a well-defined function ω:M→R\omega:M\to\mathbb{R} given by ω(p)=ωα(φα(p))\omega(p)=\omega_\alpha(\varphi_\alpha(p)) for any chart with p∈Uαp\in U_\alpha, and we identify smooth differential 00-forms on MM with such functions, called \textbf{smooth functions} on MM.

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  • Existence of Smooth Bump Functions on Euclidean Space

    lemmalem:smooth-bump-function-euclidean-2026aAnalysisMultivariable Calculus
    Let n∈n\in \reftext{def:natural-numbers-2026a}{N\mathbb{N}}, let x0x_0 be a point of \reftext{def:euclidean-space-rn-2026a}{Euclidean space} Rn\mathbb{R}^n, and let r,s∈Rr,s\in\mathbb{R} with 0<r<s0<r<s. Then there exists a \reftext{def:smooth-map-euclidean-open-set-2026a}{smooth map} χ:Rn→R\chi:\mathbb{R}^n\to\mathbb{R} such that, with dd denoting the \reftext{def:euclidean-distance-rn-2026a}{Euclidean distance} on Rn\mathbb{R}^n: 0≤χ(x)≤10\le\chi(x)\le 1 for every x∈Rnx\in\mathbb{R}^n; χ(x)=1\chi(x)=1 for every x∈Rnx\in\mathbb{R}^n with d(x,x0)≤rd(x,x_0)\le r; χ(x)=0\chi(x)=0 for every x∈Rnx\in\mathbb{R}^n with d(x,x0)≥sd(x,x_0)\ge s.

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  • Pullback Invariance of the Integral under Orientation-Preserving Smooth Diffeomorphisms

    theoremthm:pullback-invariance-integral-diffeomorphism-euclidean-2026aAnalysisGeometryMultivariable Calculus
    Let n∈n\in \reftext{def:natural-numbers-2026a}{N\mathbb{N}}, let Ω,Ω′\Omega,\Omega' be admissible domains in \reftext{def:euclidean-space-rn-2026a}{Euclidean space} Rn\mathbb{R}^n in the sense of \ref{def:continuous-n-form-support-euclidean-domain-2026a}, and let F:Ω→Ω′F:\Omega\to\Omega' be an orientation-preserving \reftext{def:smooth-diffeomorphism-euclidean-half-space-domain-2026a}{smooth diffeomorphism}. Let ω\omega be a continuous differential nn-form on Ω′\Omega' that is compactly supported in Ω′\Omega', in the sense of \ref{def:continuous-n-form-support-euclidean-domain-2026a}. Then the \reftext{def:smooth-diffeomorphism-euclidean-half-space-domain-2026a}{pullback} F∗ωF^{*}\omega is a continuous differential nn-form on Ω\Omega that is compactly supported in Ω\Omega, and ∫ΩF∗ω=∫Ω′ω,\int_{\Omega}F^{*}\omega=\int_{\Omega'}\omega, where both sides are integrals in the sense of \ref{def:integral-compactly-supported-n-form-euclidean-2026a}.

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  • Pullback by a Smooth Map Commutes with the Exterior Derivative

    theoremthm:pullback-commutes-exterior-derivative-euclidean-2026aGeometryMultivariable Calculus
    Let n,m∈n,m\in \reftext{def:natural-numbers-2026a}{N\mathbb{N}} and k∈N∪{0}k\in\mathbb{N}\cup\{0\}. Let U⊆RnU\subseteq\mathbb{R}^n and V⊆RmV\subseteq\mathbb{R}^m be \reftext{def:open-subset-euclidean-space-2026a}{open} subsets of \reftext{def:euclidean-space-rn-2026a}{Euclidean space}, let F:U→VF:U\to V be a \reftext{def:smooth-map-euclidean-open-set-2026a}{smooth map}, and let ω\omega be a \reftext{def:c1-differential-k-form-euclidean-open-set-2026b}{C1C^1 differential kk-form} on VV. Then the \reftext{def:pullback-differential-form-c1-euclidean-2026a}{pullback} F∗ωF^{*}\omega is a C1C^1 differential kk-form on UU, and d(F∗ω)=F∗(dω),d(F^{*}\omega)=F^{*}(d\omega), where on the left dd denotes the \reftext{def:exterior-derivative-c1-differential-form-euclidean-open-set-2026b}{exterior derivative} on UU applied to F∗ωF^{*}\omega, and on the right dωd\omega is the exterior derivative of ω\omega on VV, whose pullback under FF is again taken in the sense of \ref{def:pullback-differential-form-c1-euclidean-2026a}.

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  • Smooth Diffeomorphism of Euclidean or Half-Space Domains, Jacobian, and Pullback

    definitiondef:smooth-diffeomorphism-euclidean-half-space-domain-2026bGeometryMultivariable Calculus
    Let n∈n\in \reftext{def:natural-numbers-2026a}{N\mathbb{N}} and let Ω,Ω′\Omega,\Omega' be admissible domains in \reftext{def:euclidean-space-rn-2026a}{Euclidean space} Rn\mathbb{R}^n in the sense of \ref{def:continuous-n-form-support-euclidean-domain-2026a}. A \textbf{smooth diffeomorphism} F:Ω→Ω′F:\Omega\to\Omega' is a \reftext{def:bijection-sets-2026a}{bijection} with the following two properties. For every x∈Ωx\in\Omega there exist \reftext{def:open-subset-euclidean-space-2026a}{open} sets W,W′⊆RnW,W'\subseteq\mathbb{R}^n with x∈Wx\in W and F(x)∈W′F(x)\in W', and a \reftext{def:smooth-map-euclidean-open-set-2026a}{smooth map} G:W→W′G:W\to W' such that G(y)=F(y)G(y)=F(y) for every y∈W∩Ωy\in W\cap\Omega. The analogous local smooth extension condition holds for the inverse map F−1:Ω′→ΩF^{-1}:\Omega'\to\Omega at every point of Ω′\Omega'. For x∈Ωx\in\Omega, the \textbf{Jacobian matrix} JF(x)J_F(x) of FF at xx is defined to be the \reftext{def:differentiable-map-at-point-euclidean-2026a}{Jacobian matrix} JG(x)J_G(x) of any local smooth extension GG as in property 1. This does not depend on the chosen extension: any two such extensions agree on the intersection of their domains with Ω\Omega, which contains a set open in the ambient set of Ω\Omega around xx. Explicitly, if xx is interior to Ω\Omega in Rn\mathbb{R}^n that shared set contains an open ball around xx, on which equal maps have equal partial derivatives; and if Ω\Omega has ambient set the \reftext{def:closed-upper-half-space-euclidean-2026a}{closed upper half-space} and xx lies in its boundary hyperplane, the shared set contains a half-ball around xx, the first-order \reftext{def:partial-derivative-coordinate-map-2026a}{partial derivatives} of each smooth extension exist two-sidedly and are continuous, and agreement on the half-ball determines their values at xx (at interior points of the half-ball directly, and at xx by continuity); hence the Jacobian matrices of any two extensions coincide at xx. We say that FF is \textbf{orientation-preserving} if for every x∈Ωx\in\Omega the \reftext{def:determinant-real-square-matrix-2026a}{determinant} of JF(x)J_F(x) satisfies det⁡JF(x)>0.\det J_F(x)>0. Finally, let k∈N∪{0}k\in\mathbb{N}\cup\{0\}. For k≥1k\ge 1, let ω\omega be an assignment which to each point x′∈Ω′x'\in\Omega' assigns an \reftext{def:alternating-k-linear-form-euclidean-2026a}{alternating kk-linear form} ωx′\omega_{x'} on Rn\mathbb{R}^n. The \textbf{pullback} F∗ωF^{*}\omega is the assignment which to each x∈Ωx\in\Omega assigns the alternating kk-linear form given by (F∗ω)x(v1,…,vk)=ωF(x)(JF(x)v1,…,JF(x)vk)(F^{*}\omega)_x(v_1,\dots,v_k)=\omega_{F(x)}\bigl(J_F(x)v_1,\dots,J_F(x)v_k\bigr) for all vectors v1,…,vk∈Rnv_1,\dots,v_k\in\mathbb{R}^n, where JF(x)vrJ_F(x)v_r is the \reftext{def:matrix-vector-product-2026a}{matrix-vector product}. For k=0k=0 we use the convention, consistent with \ref{def:differential-k-form-euclidean-open-set-2026a}, that an assignment of alternating 00-linear forms on Ω′\Omega' is a real-valued function ω:Ω′→R\omega:\Omega'\to\mathbb{R}, and the pullback is defined by (F∗ω)(x)=ω(F(x))(F^{*}\omega)(x)=\omega(F(x)) for every x∈Ωx\in\Omega. When Ω\Omega and Ω′\Omega' are open in Rn\mathbb{R}^n, both cases agree exactly with the pullback from \ref{def:pullback-differential-form-c1-euclidean-2026a}, whose statement covers k=0k=0 by the same convention.

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  • Integral of a Compactly Supported Continuous n-Form on a Euclidean or Half-Space Domain

    definitiondef:integral-compactly-supported-n-form-euclidean-2026aAnalysisMultivariable Calculus
    Let n∈n\in \reftext{def:natural-numbers-2026a}{N\mathbb{N}}, let Ω\Omega be an admissible domain in \reftext{def:euclidean-space-rn-2026a}{Euclidean space} Rn\mathbb{R}^n with ambient set DD in the sense of \ref{def:continuous-n-form-support-euclidean-domain-2026a}, and let ω\omega be a continuous differential nn-form on Ω\Omega that is compactly supported in Ω\Omega in the sense of that definition. By \ref{lem:zero-extension-box-integral-euclidean-2026a} there exists a \reftext{def:closed-box-rn-2026a}{closed box} B⊆DB\subseteq D containing supp⁡ω\operatorname{supp}\omega, the iterated one-dimensional Riemann integrals of the zero extension f~\tilde f of the coefficient function of ω\omega over any such box exist, and the resulting value is independent of the chosen box. The \textbf{integral of ω\omega over Ω\Omega}, denoted ∫Ωω,\int_{\Omega}\omega, is defined to be this common value.

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  • Zero Extension Continuity and Box Independence of the Iterated Integral

    lemmalem:zero-extension-box-integral-euclidean-2026aAnalysisMultivariable Calculus
    Let n∈n\in \reftext{def:natural-numbers-2026a}{N\mathbb{N}}, let Ω\Omega be an admissible domain in \reftext{def:euclidean-space-rn-2026a}{Euclidean space} Rn\mathbb{R}^n with ambient set DD in the sense of \ref{def:continuous-n-form-support-euclidean-domain-2026a}, and let ω\omega be a continuous differential nn-form on Ω\Omega that is compactly supported in Ω\Omega, with coefficient function ff and zero extension f~\tilde f, all in the sense of that definition. Then the following hold. The function f~:D→R\tilde f:D\to\mathbb{R} is \reftext{def:continuous-map-at-point-euclidean-2026a}{continuous at every point} of DD. There exists a \reftext{def:closed-box-rn-2026a}{closed box} B⊆DB\subseteq D with supp⁡ω⊆B\operatorname{supp}\omega\subseteq B. When D=HnD=H^n is the \reftext{def:closed-upper-half-space-euclidean-2026a}{closed upper half-space}, such a box has the form B=[a1,b1]×⋯×[an,bn]B=[a_1,b_1]\times\cdots\times[a_n,b_n] with an≥0a_n\ge 0. Let B=[a1,b1]×⋯×[an,bn]⊆DB=[a_1,b_1]\times\cdots\times[a_n,b_n]\subseteq D be any closed box with supp⁡ω⊆B\operatorname{supp}\omega\subseteq B, where aj,bj∈Ra_j,b_j\in\mathbb{R} with aj≤bja_j\le b_j for each j∈{1,…,n}j\in\{1,\dots,n\}. For fixed (t2,…,tn)∈[a2,b2]×⋯×[an,bn](t_2,\dots,t_n)\in[a_2,b_2]\times\cdots\times[a_n,b_n] define G1(t2,…,tn)=∫a1b1f~(t1,t2,…,tn) dt1,G_1(t_2,\dots,t_n)=\int_{a_1}^{b_1}\tilde f(t_1,t_2,\dots,t_n)\,dt_1, as a one-dimensional \reftext{def:riemann-integrable-closed-interval-c54-2026b}{Riemann integral}; recursively, for r∈{2,…,n}r\in\{2,\dots,n\} and fixed (tr+1,…,tn)∈[ar+1,br+1]×⋯×[an,bn](t_{r+1},\dots,t_n)\in[a_{r+1},b_{r+1}]\times\cdots\times[a_n,b_n] define Gr(tr+1,…,tn)=∫arbrGr−1(tr,…,tn) dtr.G_r(t_{r+1},\dots,t_n)=\int_{a_r}^{b_r}G_{r-1}(t_r,\dots,t_n)\,dt_r. Then at every stage the integrand is a continuous function of the integration variable on the corresponding closed interval, hence \reftext{lem:continuous-implies-riemann-integrable-c54-2026b}{Riemann integrable}, each function GrG_r is continuous in its remaining variables, and the final value Gn∈RG_n\in\mathbb{R} is well defined. The value GnG_n obtained in claim 3 is the same for every closed box B⊆DB\subseteq D with supp⁡ω⊆B\operatorname{supp}\omega\subseteq B.

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  • Continuous n-Form, Support, and Zero Extension on a Euclidean or Half-Space Domain

    definitiondef:continuous-n-form-support-euclidean-domain-2026aAnalysisMultivariable Calculus
    Let n∈n\in \reftext{def:natural-numbers-2026a}{N\mathbb{N}}. We call a subset Ω\Omega of \reftext{def:euclidean-space-rn-2026a}{Euclidean space} Rn\mathbb{R}^n an \textbf{admissible domain} if either Ω\Omega is an \reftext{def:open-subset-euclidean-space-2026a}{open subset} of Rn\mathbb{R}^n, or Ω\Omega is a subset of the \reftext{def:closed-upper-half-space-euclidean-2026a}{closed upper half-space} HnH^n that is open in HnH^n in the sense of that definition. In the first case set D=RnD=\mathbb{R}^n, and otherwise set D=HnD=H^n; we call DD the ambient set of Ω\Omega. A differential nn-form on Ω\Omega is an assignment ω\omega which to each point x∈Ωx\in\Omega assigns an \reftext{def:alternating-k-linear-form-euclidean-2026a}{alternating nn-linear form} ωx\omega_x on Rn\mathbb{R}^n. For each i∈{1,…,n}i\in\{1,\dots,n\} let ei∈Rne_i\in\mathbb{R}^n denote the iith standard basis vector, whose iith coordinate is 11 and whose other coordinates are 00. The \textbf{coefficient function} of ω\omega is the function f:Ω→R,f(x)=ωx(e1,…,en).f:\Omega\to\mathbb{R},\qquad f(x)=\omega_x(e_1,\dots,e_n). We say that ω\omega is \textbf{continuous} if ff is \reftext{def:continuous-map-at-point-euclidean-2026a}{continuous at every point} of Ω\Omega. When Ω\Omega is open in Rn\mathbb{R}^n, this agrees with \ref{def:continuous-differential-k-form-euclidean-open-set-2026b}, because by \ref{thm:coordinate-expansion-differential-forms-euclidean-2026b} the function ff is exactly the coefficient of the top-degree basis form in the coordinate expansion of ω\omega. The \textbf{support} of ω\omega, denoted supp⁡ω\operatorname{supp}\omega, is the set of all x∈Dx\in D such that every \reftext{def:open-subset-euclidean-space-2026a}{open subset} W⊆RnW\subseteq\mathbb{R}^n with x∈Wx\in W contains a point y∈W∩Ωy\in W\cap\Omega with ωy≠0\omega_y\ne 0. We say that ω\omega is \textbf{compactly supported in Ω\Omega} if supp⁡ω\operatorname{supp}\omega is a \reftext{def:compact-space-and-subset-2026a}{compact} subset of Rn\mathbb{R}^n and supp⁡ω⊆Ω\operatorname{supp}\omega\subseteq\Omega. The \textbf{zero extension} of the coefficient function of ω\omega is the function f~:D→R\tilde f:D\to\mathbb{R} defined by f~(x)=f(x)\tilde f(x)=f(x) for x∈Ωx\in\Omega and f~(x)=0\tilde f(x)=0 for x∈D∖Ωx\in D\setminus\Omega.

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  • Stokes Theorem for Compact Oriented Smooth Manifolds with Boundary

    theoremthm:stokes-smooth-manifold-boundary-2026aAnalysisGeometryTopologyMultivariable Calculus
    Let n∈n\in \reftext{def:natural-numbers-2026a}{N\mathbb{N}} with n≥2n\ge 2, and let MM be an \reftext{def:oriented-smooth-manifold-boundary-2026a}{oriented smooth manifold with boundary} of dimension nn that is compact as defined in \ref{def:smooth-manifold-with-boundary-2026a}. Let ω\omega be a \reftext{def:smooth-differential-k-form-manifold-boundary-2026a}{smooth differential (n−1)(n-1)-form} on MM and let dωd\omega denote the \reftext{def:exterior-derivative-smooth-form-manifold-boundary-2026a}{exterior derivative} of ω\omega. Let ∂M\partial M be the \reftext{def:boundary-smooth-manifold-with-boundary-2026a}{boundary} of MM, regarded as a compact oriented smooth manifold (without boundary) of dimension n−1n-1 with the \reftext{def:induced-orientation-boundary-manifold-2026a}{induced boundary orientation}, and let ι∗ω\iota^{*}\omega denote the \reftext{def:restriction-form-boundary-manifold-2026a}{restriction} of ω\omega to ∂M\partial M where ι:∂M→M\iota:\partial M\to M is the inclusion map. Then ∫Mdω  =  ∫∂Mι∗ω,\int_{M} d\omega \;=\; \int_{\partial M} \iota^{*}\omega, where both sides are \reftext{def:integral-form-oriented-manifold-boundary-2026a}{integrals of smooth top-degree forms over compact oriented smooth manifolds}. If ∂M=∅\partial M=\varnothing, then the right-hand side is understood to be 00.

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  • Adjugate Formula for the Matrix Inverse

    theoremthm:adjugate-formula-matrix-inverse-2026b
    Let nn be a \reftext{def:natural-numbers-2026a}{natural number} with n≥2n\ge 2, and let AA be an \reftext{def:inverse-matrix-invertible-real-square-matrix-2026a}{invertible} n×nn\times n real matrix with \reftext{def:determinant-real-square-matrix-2026a}{determinant} det⁡A≠0\det A\ne 0. Then A−1=1det⁡Aadj⁡(A),A^{-1}=\frac{1}{\det A}\operatorname{adj}(A), where adj⁡(A)\operatorname{adj}(A) is the \reftext{def:minor-cofactor-adjugate-real-square-matrix-2026b}{adjugate} of AA. Equivalently, for all i,j∈{1,…,n}i,j\in\{1,\dots,n\}, (A−1)ij=Cji(A)det⁡A,(A^{-1})_{ij}=\frac{C_{ji}(A)}{\det A}, where Cji(A)C_{ji}(A) is the (j,i)(j,i) \reftext{def:minor-cofactor-adjugate-real-square-matrix-2026b}{cofactor} of AA. In particular, each entry of A−1A^{-1} is a polynomial in the entries of AA divided by det⁡A\det A.

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    Authors Claude-Sonnet-4-6, Aaron · Created

  • Let k,nk,n be \reftext{def:natural-numbers-2026a}{natural numbers} with k≥1k\ge 1, let UU be an \reftext{def:open-subset-euclidean-space-2026a}{open} subset of \reftext{def:euclidean-space-rn-2026a}{Euclidean space} Rn\mathbb{R}^n, and let f,g:U→Rf,g:U\to\mathbb{R} be maps of class \reftext{def:ck-map-euclidean-open-set-2026b}{CkC^k} on UU. (i) The pointwise product (fg):U→R(fg):U\to\mathbb{R}, defined by (fg)(x)=f(x) g(x)(fg)(x)=f(x)\,g(x) for x∈Ux\in U, is of class \reftext{def:ck-map-euclidean-open-set-2026b}{CkC^k} on UU. (ii) If g(x)≠0g(x)\ne 0 for every x∈Ux\in U, then the pointwise quotient f/g:U→Rf/g:U\to\mathbb{R}, defined by (f/g)(x)=f(x)/g(x)(f/g)(x)=f(x)/g(x) for x∈Ux\in U, is of class \reftext{def:ck-map-euclidean-open-set-2026b}{CkC^k} on UU.

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  • Minor, Cofactor, and Adjugate of a Real Square Matrix

    definitiondef:minor-cofactor-adjugate-real-square-matrix-2026b
    Let nn be a \reftext{def:natural-numbers-2026a}{natural number} with n≥2n\ge 2, and let A=(aij)1≤i,j≤nA=(a_{ij})_{1\le i,j\le n} be an n×nn\times n real matrix. For indices i,j∈{1,…,n}i,j\in\{1,\dots,n\}, the (i,j)(i,j) \textit{minor} of AA, denoted Mij(A)M_{ij}(A), is the \reftext{def:determinant-real-square-matrix-2026a}{determinant} of the (n−1)×(n−1)(n-1)\times(n-1) real matrix obtained from AA by deleting row ii and column jj. The (i,j)(i,j) \textit{cofactor} of AA is Cij(A)=(−1)i+jMij(A).C_{ij}(A)=(-1)^{i+j}M_{ij}(A). The \textit{adjugate} matrix of AA (also called the classical adjoint) is the n×nn\times n real matrix adj⁡(A)=(Cji(A))1≤i,j≤n,\operatorname{adj}(A)=\bigl(C_{ji}(A)\bigr)_{1\le i,j\le n}, i.e., the matrix whose (i,j)(i,j) entry is the (j,i)(j,i) cofactor of AA.

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  • CkC^k Map on an Open Subset of Euclidean Space

    definitiondef:ck-map-euclidean-open-set-2026b
    Let k,n,mk,n,m be \reftext{def:natural-numbers-2026a}{natural numbers} with k≥1k\ge 1. Let UU be an \reftext{def:open-subset-euclidean-space-2026a}{open} subset of \reftext{def:euclidean-space-rn-2026a}{Euclidean space} Rn\mathbb{R}^n, and let f=(f1,…,fm):U→Rmf=(f_1,\dots,f_m):U\to\mathbb{R}^m. We say that ff is of class CkC^k on UU if for every multi-index α\alpha of length nn with \reftext{def:order-factorial-multi-index-2026a}{order} ∣α∣≤k|\alpha|\le k and every index j∈{1,…,m}j\in\{1,\dots,m\}, the partial derivative ∂αfj\partial^\alpha f_j of \reftext{def:partial-derivative-order-alpha-2026a}{order α\alpha} exists on UU, and the resulting function ∂αfj:U→R\partial^\alpha f_j:U\to\mathbb{R} is \reftext{def:continuous-map-at-point-euclidean-2026a}{continuous at every point of UU}. A real-valued map f:U→Rf:U\to\mathbb{R} is of class CkC^k if it is of class CkC^k as a map from UU to R1\mathbb{R}^1.

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