Ivancevic_Applied-Diff-Geom

176 Applied Differential Geometry: A Modern IntroductionClearly, when M is itself a 1D manifold, (3.38) gives precisely the ordinaryintegration of a function α(x) over x, so the above notation is indeednatural.The 1−forms on M are part of an algebra, called the exterior algebra,or Grassmann algebra on M. The multiplication ∧ in this algebra is calledwedge product (see (3.40) below), and it is skew–symmetric,dx i ∧ dx j = −dx j ∧ dx i .One consequence of this is that dx i ∧ dx i = 0.3.6.2.2 k−Forms on MA differential form, or an exterior form α of degree k, or a k−form for short,is a section of the vector bundle Λ k T ∗ M, i.e., α : M → Λ k T ∗ M. In otherwords, α(m) : T m M × ... × T m M → R (with k factors T m M) is a functionthat assigns to each point m ∈ M a skew–symmetric k−multilinear map onthe tangent space T m M to M at m. Without the skew–symmetry assumption,α would be called a (0, k)−tensor–field. The space of all k−forms isdenoted by Ω k (M). It may also be viewed as the space of all skew symmetric(0, k)−tensor–fields, the space of all mapsΦ : X k (M) × ... × X k (M) → C k (M, R),which are k−linear and skew–symmetric (see (3.40) below). We putΩ k (M) = C k (M, R).In particular, a 2−form ω on an n−manifold M is a section of the vectorbundle(Λ 2 T ∗ M. If (U, φ) is a chart at a point m ∈ M with local coordinatesx 1 , ..., x n) let {e 1 , ..., e n } = {∂ x 1, ..., ∂ x n} – be the corresponding basis forT m M, and let { e 1 , ..., e n} = { dx 1 , ..., dx n} – be the dual basis for TmM.∗Then at each point m ∈ M, we can write a 2−form ω asω m (v, u) = ω ij (m) v i u j , where ω ij (m) = ω m (∂ x i, ∂ x j ).Similarly to the case of a 1–form α (3.37), one would like to define a2–form ω as something which can naturally be integrated over a 2D surfaceΣ within a smooth manifold M. At a specific point x ∈ M, the tangentplane to such a surface is spanned by a pair of tangent vectors, (ẋ 1 , ẋ 2 ). So,to generalize the construction of a 1–form, we should give a bilinear mapfrom such a pair to R. The most general form of such a map isω ij (x) dx i ⊗ dx j , (3.39)