Download Leçons sur les systemes orthogonaux et les coordonnees by Gaston Darboux PDF

By Gaston Darboux

Excerpt from Leçons sur les Systèmes Orthogonaux Et les Coordonnées Curvilignes

Dans les Leçons sur los angeles théorie des surfaces, j avais déjà fait connaître, d'une manière incidente, différentes professional priétés des systèmes triples orthogonaux et des coordon nées curvilignes; mais j'avais réservé le développement régulier et systématique des théories qui se rattachent à ce beau sujet pour le nouveau Traité dont je begin aujourd'hui l. a. publication.

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Extra info for Leçons sur les systemes orthogonaux et les coordonnees curvilignes

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48 3. 9) classify global solutions, somewhat like in minimal surface theory. We know that F ∗ (u) is, in particular a set of finite perimeter and therefore almost every point, with respect to H n−1 F ∗ (u), is a differentiability point. That is, if x is one of those points, at x is well defined a normal vector ν = ν(x) such that, if we set + Ω+ r = {y : r(y − x) ∈ Ω (u)} P + = P + (x, ν) = {y : y − x, ν > 0} π = π(x, ν) = {y : y − x, ν = 0} and let B = B(x) a (small) ball centered at x, then Per(Ω+ r ∩ B) converges in the sense of vector measures to Per(P + ∩ B), that is, for any continuous vector field ϕ: r→0 ∂Ω+ r ∩B ϕ, ν d Per −−−→ ϕ, ν dH n−1 .

U Bε (xj ) therefore the quantities b) and c) are comparable. 3. STRONG RESULTS 47 since, for proper choices of c we can make Ncε (F (u)) ∩ BR ⊂ {0 < u < ε} ∩ BR or vice versa. It follows that the quantities a), b) and c) are all comparable to Rn−1 . Finally, let {Brj (xj )}, xj ∈ F (u), a finite covering of F (u)∩BR by balls of radius rj < ε, that approximates H n−1 (F (u) ∩ BR ). Let r < min rj and {Br (xkj )} a finite overlapping covering for F (u) ∩ Brj (xj ). Then, on one hand |∂Br (xkj )| ≤ cRn−1 k,j by the argument above with ε = r.

Let y ∈ Bε (x) and notice that if τ ∈ Γ( θ2 , en ) and τ¯ = τ − (y − x) τ − τ | = |x − y| ≤ |τ | sin θ2 . Also then α(τ, τ¯) ≤ 2θ , since |¯ |¯ τ | ≥ |τ | − |τ | sin since θ 2 1 θ ≥ |τ | 2 2 < π4 . 8, we deduce that inf B1/8 (x0 ) Dτ¯ u ≥ c0 ν, τ¯ |∇u(x0 )| ≥ c ν, τ¯ u(x0 ) τ | cos α(ν, τ¯) ≥ c1 |¯ sup u B1/8 (x0 ) ≥ bε sup u B1/8 (x0 ) where b = b(τ ) = C cos( θ2 + α(ν, τ )). 5 are satisfied. 6. 4, perhaps with a slightly different enlarged cone, that we still denote by Γ(θ¯1 , ν¯1 ). 4. 3, with θ = θ/2, θ θ 1 + cμ cos + α(ν, τ ) 2 2 θ [1 + cμ sin E(τ )] = |τ | sin 2 θ +μ ¯E(τ ≡ ρ(τ ) ≥ |τ | sin 2 (1 + bμ)ε = |τ | sin with μ ¯ = μc θ20 .

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