Capillarity and wetting phenomena : drops, bubbles, pearls, by Pierre-Gilles de Gennes

By Pierre-Gilles de Gennes

The examine of capillarity is in the middle of a veritable explosion. what's provided this is now not a complete evaluate of the most recent study yet particularly a compendium of rules designed for the undergraduate scholar and for readers drawn to the physics underlying those phenomena.

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Otherwise the manifold is called non-orientable. 23 Let M1 and M2 be two differentiable manifolds of dimension l and m, respectively. 56) is differentiable in x−1 (p) (Fig. 21). The map g is differentiable in an open subset of M1 if it is differentiable at every point of the subset. x–1(p) y–1 ˚ g ˚ x U (U ) x –1 g y ˚ ˚ V p x(U) g M1 g(p) y(V ) Fig. 7 Geometric and kinematic foundations of Lagrangian mechanics 39 Note that by choosing M2 = R this defines the notion of a differentiable map (in an obvious way we can also define the notion of a map of class Ck or C∞ ) from M to R.

U˙ l ). It is then natural to consider the velocity vectors corresponding to the l-tuples (1, 0, . . , 0), (0, 1, . . , 0), . . , (0, 0, . . , 1). 58) exactly as in the case of a regular l-dimensional submanifold. It is now easy to show that for p ∈ M and v ∈ Tp M , it is possible to find a curve γ : (−ε, ε) → M such that γ(0) = p and γ(0) ˙ = v. Indeed, it is enough to consider the decomposition l vi v= i=1 ∂x (0) ∂ui for some local parametrisation (U, x), and to construct a map µ : (−ε, ε) → U such that its components ui (t) have derivatives ui (0) = vi .

26 Let g : M1 → M2 be a differentiable map between the differentiable manifolds M1 , M2 of dimension l, m, respectively. The linear map which ˙ associates w ∈ Tg(p) M2 , defined by with every v ∈ Tp M1 , defined by v = γ(0), ˙ w = β(0), with β = g ◦ γ, is the differential dgp : Tp M1 → Tg(p) M2 . We showed that the map dgp acts on the components of the vectors in Tp M1 as the row-by-column product with the Jacobian matrix ∂(f1 , . . , fm )/∂(u1 , . . , ul ). 7 Geometric and kinematic foundations of Lagrangian mechanics 41 This happens in particular when the map is the change of parametrisation on a manifold (the Jacobian is in this case a square matrix).

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