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Dover e-book: artwork Nouveau Motifs ДИЗАЙН,ДИЗАЙН и ГРАФИКА Название: paintings Nouveau Motifs Автор: Dover guides Год: 2002 Размер: 31,6 MB Качество: отличное Серия или Выпуск: digital Clip eighty five

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Xn ) denote a continuous vectorfield and n ω= ai (x) dxi i=1 the associate 1-form or Pfaffian form. Furthermore, let X(t) = x1 (t), . . , xn (t) : T → Rn ∈ C 1 (T ) represent a regular C 1 -curve defined on the parameter interval T = (a, b). Then we observe b n ai X(t) xi (t) ω= X a dt. i=1 We shall investigate curvilinear integrals in Section 6 more intensively. 16. (Surface integrals) Let the continuous vector-field a(x) = with the associate (n − 1)-form a1 (x1 , . . , xn ), . . , an (x1 , . .

5. Now we consider a specific (n − 1)-form in Rn , namely n ai (x)(−1)i+1 dx1 ∧ . . ∧ dxi−1 ∧ dxi+1 ∧ . . ∧ dxn , ω= i=1 whose exterior derivative takes on the following form: n (−1)i+1 dai (x) ∧ dx1 ∧ . . ∧ dxi−1 ∧ dxi+1 ∧ . . ∧ dxn dω = i=1 n (−1)i+1 = i,j=1 n (−1)i+1 = i=1 n = i=1 ∂ai (x) dxj ∧ dx1 ∧ . . ∧ dxi−1 ∧ dxi+1 ∧ . . ∧ dxn ∂xj ∂ai (x) dxi ∧ dx1 ∧ . . ∧ dxi−1 ∧ dxi+1 ∧ . . ∧ dxn ∂xi ∂ai (x) ∂xi dx1 ∧ . . ∧ dxn = div a(x) dx1 ∧ . . ∧ dxn . 6. The vector-field a(x) = a1 (x), . .

D. 11. (Chain rule for differential forms) Let ω denote a continuous m-form in an open set O ⊂ Rn . Furthermore, we consider the open sets T ⊂ Rl and T ⊂ Rl - with l , l ∈ N - where the C 1 -functions Φ, Ψ are defined due to Ψ : T →T , Φ : T →O Ψ Φ z −→ y −→ x. with Then the following identity holds true: (ωΦ )Ψ = ωΦ◦Ψ . im Φ ◦ Ψ (z) d(ϕi1 ◦ Ψ ) ∧ . . ,jm ∧ ... ∧ ∧ ... im Φ(y) dϕi1 ∧ . . ,im ∂ϕim ∂ψjm dzkm ∂yjm ∂zkm ∂ϕim dψjm ∂yjm , y=Ψ (z) and consequently ωΦ◦Ψ = (ωΦ )Ψ . Here we perform our summation over the indices i1 , .

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Art Nouveau Motifs

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