Stokes’ Theorem
نویسنده
چکیده
I. Introduction. Stokes' theorem on a manifold is a central theorem of mathematics. Special cases include the integral theorems of vector analysis and the Cauchy-Goursat theorem. My purpose here is to prove this version of Stokes' Theorem. Let ω be a continuous differential (n − 1)-form on a compact oriented n-manifold M with boundary ∂M. Suppose that ω is differen-tiable on M − ∂M and dω is Lebesgue integrable there. Then
منابع مشابه
On the L–Stokes theorem and Hodge theory for singular algebraic varieties
We discuss aspects of the L–Stokes theorem on certain manifolds with singularities. We show that the L–Stokes theorem does not hold on real projective varietes, even for isolated singularities. For a complex projective variety of complex dimension n, with isolated singularities, we show that the Laplacians of the de Rham and Dolbeault complexes are discrete operators except possibly in degrees ...
متن کامل7 M ay 2 00 2 On the L 2 – Stokes theorem and Hodge theory for singular algebraic varieties
For a projective algebraic variety V with isolated singularities, endowed with a metric induced from an embedding, we consider the analysis of the natural partial differential operators on the regular part of V . We show that, in the complex case, the Laplacians of the de Rham and Dolbeault complexes are discrete operators except possibly in degrees n, n±1, where n is the complex dimension of V...
متن کاملar X iv : m at h - ph / 0 01 20 35 v 1 1 9 D ec 2 00 0 Non - Abelian Stokes theorem in action
In this short review main issues related to the non-Abelian Stokes theorem have been addressed. The two principal approaches to the non-Abelian Stokes theorem, operator and two variants (coherent-state and holomorphic) of the path-integral one, have been formulated in their simplest possible forms. A recent generalization for a knotted loop as well as a suggestion concerning higher-degree forms...
متن کاملA generalized “surfaceless” Stokes’ theorem
We derive a generalized Stokes' theorem, valid in any dimension and for arbitrary loops, even if self intersecting or knotted. The generalized theorem does not involve an auxiliary surface, but inherits a higher rank gauge symmetry from the invariance under deformations of the surface used in the conventional formulation.
متن کاملIntroduction to Calculus in Several Variables
1 2 Contents 0. One-variable calculus 1. The derivative 2. Inverse function and implicit function theorem 3. Fundamental local existence theorem for ODE 4. The Riemann integral in n variables 5. Integration on surfaces 6. Differential forms 7. Products and exterior derivatives of forms 8. The general Stokes formula 9. The classical Gauss, Green, and Stokes formulas 10. Holomorphic functions and...
متن کامل