Secondary Calculus and the Covariant Phase Space

Vector calculus for tamed Dirichlet spaces

In the language of $L^\infty$-modules proposed by Gigli, we introduce a first order calculus on a topological Lusin measure space $(M,\mathfrak{m})$ carrying a quasi-regular, strongly local Dirichlet form $\mathscr{E}$. Furthermore, we develop a second order calculus if $(M,\mathscr{E},\mathfrak{m})$ is tamed by a signed measure in the extended Kato class in the sense of Erbar, Rigoni, Sturm and Tamanini. This allows us to define e.g. Hessians, covariant and exterior derivatives, Ricci curvature, and second fundamental form.

Covariant Derivatives of Mutivector and Extensor Fields

We give in this paper which is the fifth in a series of eight a theory of covariant derivatives of multivector and extensor fields based on the geometric calculus of an arbitrary smooth manifold M, and the notion of a connection extensor field defining a parallelism structure on M. Also we give a novel and intrinsic presentation (i.e., one that does not depend on a chosen orthonormal moving frame) of the torsion and curvature fields of Cartan's theory. Two kinds of Cartan's connection fields are identified, and both appear in the intrinsic Cartan's structure equations satisfied by the Cartan's…

Metric Compatible Covariant Derivatives

This paper, sixth in a series of eight, uses the geometric calculus on manifolds developed in previous papers of the series to introduce through the concept of a metric extensor field g a metric structure for a smooth manifold M. The associated Christoffel operators, a notable decomposition of that object and the associated Levi-Civita connection field are given. The paper introduces also the concept of a geometrical structure for a manifold M as a triple (M,g,gamma), where gamma is a connection extensor field defining a parallelism structure for M . Next, the theory of metric compatible covar…

离散外微分算术（DEC）用于表面纳维-斯托克斯方程

Discrete exterior calculus (DEC) for the surface Navier-Stokes equation

本文提出了一种基于离散外微分算术（DEC）的数值方法，用于求解不可压缩表面纳维-斯托克斯方程，展示了该方法在拓扑、几何和流体性质间的相互作用以及二阶收敛性。

Iterated Differential Forms V: C-Spectral Sequence on Infinite Jet Spaces

In the preceding note math.DG/0610917 the $Λ_{k-1}\mathcal{C}$--spectral sequence, whose first term is composed of \emph{secondary iterated differential forms}, was constructed for a generic diffiety. In this note the zero and first terms of this spectral sequence are explicitly computed for infinite jet spaces. In particular, this gives an explicit description of secondary covariant tensors on these spaces and some basic operations with them. On the basis of these results a description of the $Λ_{k-1}\mathcal{C}$--spectral sequence for infinitely prolonged PDE's will be given in the subsequen…

极小曲面：第一和第二变分的拉格朗日推导

Minimal surfaces: A Lagrangian derivation of first and second variations

本文通过拉格朗日方法，利用拉回协变导数，严格推导了极小曲面的第一和第二变分，并证明所有切向变分为零。