Homework set 8 - due 04 / 06 / 2015

• Integration of forms. I review the basics about integrating an n-form over an n-dimensional manifold. Let M be a smooth n-manifold. We need M to be orientable. There are several equivalent characterizations of orientability; for example, M is orientable if there exists a nowhere vanishing continuous n-form ω. Given such an n-form we can select from a smooth atlas of M a subatlas for which the Jacobian determinant of coordinate changes are always positive. This subatlas can be defined by selecting only coordinatesφ−1 = (x1, . . . , xn) so that ω(∂/∂x1, . . . ,∂/∂xn) > 0. With the slight abuse of notation of thinking of ∂/∂x j as vectors either tangent to M or on Rn , this is the same as saying that φ∗ω= ρd x1 ∧·· ·∧d xn where ρ is a positive real-valued function. Let {(Uα,φα) :α ∈ A} be an oriented atlas of smooth parametrizations of M , where Uα ⊂Rn and φα : Uα ⊂Rn →Vα =φα(Uα) ⊂ M is a diffeomorphism for eachα. We assume that {Vα :α ∈ A} is locally finite. This means that for each p ∈ M there is an open neighborhood of p that intersects at most finitely many Vα. Let {ηα :α ∈ A} be a smooth partition of unity subordinate to the open cover {Vα :α ∈ A}. By definition, {ηα :α ∈ A} satisfies the properties: 1. ηα : M → [0,1] has compact support contained in Vα for each α.