Stoke's Theorem as the modern working mathematician sees it. A student with a good course in calculus and linear algebra behind him should find this book 

Chapter 5. Integration and Stokes’ theorem 63 5.1. Integration of forms over chains 63 5.2. The boundary of a chain 66 5.3. Cycles and boundaries 68 5.4. Stokes’ theorem 70 Exercises 71 Chapter 6. Manifolds 75 6.1. The definition 75 6.2. The regular value theorem 82 Exercises 88 Chapter 7. Differential forms on manifolds 91 iii

Now it says that this is a "space divergence in the metric σ " and therefore ∫ S σ i k [ ∇ i σ (f k d x k)] = 0 and that the reason for this is Stokes theorem. Stokes Theorem. Stokes Theorem is also referred to as the generalized Stokes Theorem. It is a declaration about the integration of differential forms on different manifolds.

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Vector fields and flows, the Lie bracket and Lie derivative. Exterior algebra, differential forms, exterior derivative, Cartan formula in terms of Lie derivative. Orientability. Partitions of unity, integration on oriented manifolds.

In particular, your closed form $\omega$ is exact; that is, there is an $(n-2)$-form $\eta$ with $d\eta = \omega$. You can now use Stokes' theorem in the usual way, together with the fact that $\partial M = \emptyset$, to show that your integral is $0$.

In general, if M is a manifold with corners then Integration on Manifolds Stokes’ Theorem on Manifolds Theorem Stokes’ Theorem on Manifolds. If M is a compact oriented smooth k-dimensional manifold-with-boundary, and ω is a smooth (k −1) form on M, then Z M dω = Z ∂M ω. Proof. Case 1.

Lecture 14. Stokes’ Theorem In this section we will define what is meant by integration of differential forms on manifolds, and prove Stokes’ theorem, which relates this to the exterior differential operator. 14.1 Manifolds with boundary In defining integration of differential forms, it will be convenient to introduce

Manifolds In order to extend Stokes’ Theorem to higher dimensions, we must introduce the concept of manifolds, which will serve as the basic setting in which the theorem can be constructed. Even before that, however, we must rst de ne the class of linear maps that serve to describe manifolds. De nition 2.1 ([1, De nition 2.6.2]). Let Abe Idea. The Stokes theorem (also Stokes' theorem or Stokes's theorem) asserts that the integral of an exterior differential form on the boundary of an oriented manifold with boundary (or submanifold or chain of such) equals the integral of the de Rham differential of the form on the manifold itself. (The theorem also applies to exterior pseudoforms on a chain of pseudoriented submanifolds.) The general theorem of Stokes on manifolds with boundary states the deceivingly simple formula Z M d!= Z @M!; where !is a di erentiable (m 1)-form on a compact oriented m-dimensional man-ifold M. To fully understand the formula though, we need to describe all the notions it contains.

Stokes’ theorem is a generalization of the fundamental theorem of calculus. For a compact orientable «-manifold R Stokes' theorem implies that (1) [da = 0 for every differentiate (n — l)-form a on R. In case R is an open relatively compact subset of a Riemannian «-manifold Bochner [1] established (1) for (n — l)-forms a vanishing "in average" at the boundary of R with da integrable. Gaffney [4] 2. Stokes’ Theorem on Manifolds Having so far avoided all the geometry and topology of manifolds by working on Eu-clidean space, we now turn back to working on manifolds.
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Integration on manifolds, explanations of Stokes' theorem and de Rham  Basics on smooth manifolds and mappings between manifolds, tangent and cotangent space, tensors, differential forms.

Partitions of unity, integration on oriented manifolds.
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Otillbörda teoremer. Förklara! Great Farm Theorem bevisas? Navier-Stokes-ekvationerna beskriver rörelsen av en viskös vätska. En av de viktigaste typerna 

curI F dA " s + (ali _ a13) dz 1\ dx az ax + (a12 _ ali) dx 1\ dy . ax ay The argument principal, in particular, may be easily deduced fr om Green's theorem provided that you know a little about complex analytic functions. The general theorem of Stokes on manifolds with boundary states the deceivingly simple formula Z M d!= Z @M!; where !is a di erentiable (m 1)-form on a compact oriented m-dimensional man-ifold M. To fully understand the formula though, we need to describe all the notions it contains. Integration on Manifolds Stokes’ Theorem on Manifolds Theorem Stokes’ Theorem on Manifolds. If M is a compact oriented smooth k-dimensional manifold-with-boundary, and ω is a smooth (k −1) form on M, then Z M dω = Z ∂M ω.