By Alexei Kushner
Tools from touch and symplectic geometry can be utilized to unravel hugely non-trivial nonlinear partial and usual differential equations with no resorting to approximate numerical equipment or algebraic computing software program. This booklet explains how it truly is performed. It combines the readability and accessibility of a complicated textbook with the completeness of an encyclopedia. the elemental principles that Lie and Cartan built on the finish of the 19th century to rework fixing a differential equation right into a challenge in geometry or algebra are the following remodeled in a unique and glossy manner. Differential equations are regarded as part of touch and symplectic geometry, in order that the entire equipment of Hodge-deRham calculus could be utilized. during this approach a large type of equations should be tackled, together with quasi-linear equations and Monge-Ampere equations (which play a huge function in sleek theoretical physics and meteorology).
Read Online or Download Contact geometry and nonlinear differential equations PDF
Best differential geometry books
This publication supplies the fundamental notions of differential geometry, resembling the metric tensor, the Riemann curvature tensor, the elemental types of a floor, covariant derivatives, and the elemental theorem of floor concept in a self-contained and available demeanour. even supposing the sector is frequently thought of a classical one, it has lately been rejuvenated, because of the manifold purposes the place it performs an important position.
Introduces uniform structures of lots of the recognized compactifications of symmetric and in the community symmetric areas, with emphasis on their geometric and topological buildings fairly self-contained reference geared toward graduate scholars and examine mathematicians drawn to the purposes of Lie idea and illustration idea to research, quantity idea, algebraic geometry and algebraic topology
Multivariable research is a crucial topic for mathematicians, either natural and utilized. except mathematicians, we predict that physicists, mechanical engi neers, electric engineers, structures engineers, mathematical biologists, mathemati cal economists, and statisticians engaged in multivariate research will locate this booklet tremendous important.
- The Geometry of Hamilton and Lagrange Spaces
- Fractals, Wavelets, and their Applications: Contributions from the International Conference and Workshop on Fractals and Wavelets
- The mathematical works of J.H.C.Whitehead. Vol.1
- Differential topology and quantum field theory
Extra info for Contact geometry and nonlinear differential equations
We shall discuss this idea later in the context of harmonic map theory, see chapter 4. 3 Euler–Lagrange equations on Riemannian manifolds 39 Let (M, g) be a Riemannian manifold and φ ∈ F(M). 9) ∂xj ∂xj where ∇ is the Levi-Civita connection on M. Consider a map : M → N, where M and N are Riemannian manifolds. The ﬁrst is the space of parameters and the second the space of coordinates. , yn ) are coordinates around (p) ∈ N, we deﬁne the vector ﬁeld ; i ∈ X (M) by ;i = ∗ ∂ = ∂xi j ;i ∂ . 10) In the particular case when M = Rt , we obtain the tangent vector ﬁeld along ˙ (t) = ∗ d .
Such an immersion is called isometric. 19) i=1 where Y = Y i ei , X = Xi ei , and e1 = (1, 0, . . , 0), . . , en = (0, . . , 0, 1). If ∇ is the Levi-Civita connection on M, the second fundamental form of the immersion f is the two-covariant, symmetric tensor ﬁeld on M h(X, Y ) = ∇˜ X Y − ∇X Y, ∀X, Y ∈ X (M). 20) is called Gauss’s formula, and we have h(X, Y ) = nor (∇˜ X Y ), ∇X Y = tan (∇˜ X Y ), where nor (tan) represents the normal (tangential) component with respect to M. 8 The mean curvature vector ﬁeld of the submanifold M of Rn is H = 1 T raceg h.
2) where c = min V . Under certain conditions M is supposed to be a connected submanifold of E, so that the scalar product : E × E → R induces a Riemannian metric g on M. 3) 2 which will be the Lagrangian for the harmonic maps and will be considered later. In geometry the notion was introduced by J. H. Sampson, see .