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.

About the Publisher

Forgotten Books publishes thousands of infrequent and vintage books. locate extra at

This booklet is a duplicate of a major historic paintings. Forgotten Books makes use of cutting-edge know-how to digitally reconstruct the paintings, retaining the unique structure while repairing imperfections found in the elderly replica. In infrequent situations, an imperfection within the unique, reminiscent of a blemish or lacking web page, can be replicated in our variation. We do, in spite of the fact that, fix the majority of imperfections effectively; any imperfections that stay are deliberately left to maintain the country of such ancient works.

Show description

Read Online or Download Leçons sur les systemes orthogonaux et les coordonnees curvilignes PDF

Similar geometry and topology books

Low Dimensional Topology

During this quantity, that's devoted to H. Seifert, are papers in response to talks given on the Isle of Thorns convention on low dimensional topology held in 1982.

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

Sample text

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 .

Download PDF sample

Rated 4.74 of 5 – based on 18 votes