Related papers: The divergence and curl in arbitrary basis
We study the spectral properties of curl, a linear differential operator of first order acting on differential forms of appropriate degree on an odd-dimensional closed oriented Riemannian manifold. In three dimensions its eigenvalues are…
We present a mixed method for the linearized elasticity equations with independent approximation of the curl of the displacements. The curl can be seen as a drilling degree of freedom allowing for coupling with rotating objects and the…
The Laplace-Beltrami operator on (the surface of) a triaxial ellipsoid admits a sequence of real eigenvalues diverging to plus infinity. By introducing ellipsoidal coordinates, this eigenvalue problem for a partial differential operator is…
In different branches of physics, we frequently deal with vector del operator ($\vec{\nabla}$). This del operator is generally used to find curl or divergence of a vector function or gradient of a scalar function. In many important cases,…
We give detailed exposition of modern differential geometry from global coordinate independent point of view as well as local coordinate description suited for actual computations. In introduction, we consider Euclidean spaces and different…
This paper concerns the theoretical and numerical analysis of a free boundary problem for the Laplace equation, with a curvature condition on the free boundary. This boundary is described as the graph of a function, and contact angles are…
A computer-algebra aided method is carried out, for determining geometric objects associated to differential operators that satisfy the elliptic ansatz. This results in examples of Lame curves with double reduction and in the explicit…
We construct all orthogonal separating coordinates in constant curvature spaces of arbitrary signature. Further, we construct explicit transformation between orthogonal separating and flat or generalised flat coordinates, as well as…
These notes are designed for those who either plan to work in differential geometry, or at least want to have a good reason not to do it. We discuss smooth curves and surfaces -- the main gate to differential geometry. We focus on the…
We study the asymptotic behaviour of regularized determinants of certain Laplace type operators with respect to singular deformations of the underlying manifold which are obtained by stretching a tubular neighborhood of an embedded…
Neural Ordinary Differential Equations model dynamical systems with ODEs learned by neural networks. However, ODEs are fundamentally inadequate to model systems with long-range dependencies or discontinuities, which are common in…
We show that the spectrum of the curl operator on a generic smoothly bounded domain in three-dimensional Euclidean space consists of simple eigenvalues. The main new ingredient in our proof is a formula for the variation of curl eigenvalues…
This is a review of some coordinate-free calculi of pseudodifferential operators developed in the last years. As an application, we use a coordinate-free calculus to obtain new results on the behaviour of the spectral projections of a…
Using classical differential geometry, the problem of elastic curves and surfaces in the presence of long-range interactions $\Phi$, is posed. Starting from a variational principle, the balance of elastic forces and the corresponding…
Exploring the relationship between geometry and the resonant frequencies of a shape is of interest to pure and applied mathematicians. These resonant frequencies are related to the spectrum of the Laplacian, a partial differential operator.…
We review some of the geometry of the quantum projective plane with emphasis on the construction of a differential calculus and of the Dirac operator (of a spin^c-structure). We also report on anti-self-dual connections on line bundles, the…
Conservation laws, in for example, electromagnetism, solid and fluid mechanics, allow an exact discrete representation in terms of line, surface and volume integrals. We develop high order interpolants, from any basis that is a partition of…
Nonlocal operators that have appeared in a variety of physical models satisfy identities and enjoy a range of properties similar to their classical counterparts. In this paper we obtain Helmholtz-Hodge type decompositions for two-point…
We provide novel linear combination of unitaries decompositions for a class of discrete elliptic differential operators. Specifically, Poisson problems augmented with periodic, Dirichlet, Neumann, Robin, and mixed boundary conditions are…
Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. In this survey we discuss the following two fundamental Discretization Principles: the…