微分几何斯托克斯公式

微分几何斯托克斯公式
英文回答：
Stokes' Theorem in Differential Geometry.
Stokes' theorem is a fundamental theorem in
differential geometry that relates the integral of a differential form over a boundary to the integral of its exterior derivative over the region bounded by the boundary. It is a generalization of the Green-Stokes theorem in
vector calculus.
The theorem is stated as follows:
Let M be a smooth, oriented n-manifold, and let ω be a smooth (n-1)-form on M. Then.
∫∂M ω = ∫M dω。

where d is the exterior derivative.
Stokes' theorem has many important applications in differential geometry, including:
The Gauss-Bonnet theorem: This theorem relates the curvature of a surface to the integral of its Gaussian curvature.
The de Rham cohomology theorem: This theorem states that the de Rham cohomology of a smooth manifold is isomorphic to the singular cohomology of the manifold.
The Hodge decomposition theorem: This theorem states that every differential form on a smooth manifold can be decomposed into a unique sum of exact, coexact, and harmonic forms.
中文回答：
斯托克斯公式在微分几何中。

斯托克斯公式是微分几何中一个基本定理，它将流形边界的微
分形式的积分与边界所包围区域中其外导数的积分联系起来。

它是
矢量微积分中格林-斯托克斯定理的推广。

该定理表述如下：
设 M 是一个光滑的定向 n 流形，ω 是 M 上一个光滑的 (n-1) 形式。

然后。

∫∂M ω = ∫M dω。

其中 d 是外导数。

斯托克斯公式在微分几何中有许多重要的应用，包括：
高斯-博内定理，该定理将曲面的曲率与其高斯曲率的积分联
系起来。

德拉姆上同调定理，该定理指出，光滑流形上的德拉姆上同调
与该流形的奇异上同调同构。

霍奇分解定理，该定理指出，平滑流形上的每个微分形式都可
以唯一分解为精确形式、协精确形式和调和形式之和。