In this paper,the basic theory of stochastic systems of It type is summarized,including the It stochastic analysis,the definition of It stochastic differential equations,It differential formula and the theorems on existence and uniqueness of solutions of It stochastic differential equations.

These properties could be used to get low order interpolation numerical differential formula quickly.

The above conclusion is demonstrated in the light of poincare theorem, it is demonstrated by using contraction (interior product) of vector field and differential form as well as operation of exterior differentiation.

By estimating the koppelman kernel on Complex Manifolds, the difference between the koppelman kernel on complex manifolds and the Bochner Martinelli koppelman on C n was obtained;and then by utilizing the koppelman formula and the result as above, the jump formula of differential forms under Berndtsson transform on Complex manifolds was derived.

By the natural and harmonious relationship between differential forms and differential equations and between differential forms and vector analysis, we discuss the properties, which are covariant under the transformation of coordinates in the framework of differential forms, of particle motion in a central force field.

The Hypo-elliptic Differential Forms on Smooth Manifolds;

Let X be a smooth oriented Riemannian n-manifold without boundary,l-form W be WT2 class of differential forms on X.