Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 181diagram commutes:Ω k (N)d❄Ω k+1 (N)ϕ ∗ϕ ∗✲ Ω k (M)d❄✲ Ω k+1 (M)(7) Analogously, d is natural with respect to diffeomorphism ϕ ∗ = (F ∗ ) −1 ;that is, ϕ ∗ dω = dϕ ∗ ω, or the following diagram commutes:Ω k (N)d❄Ω k+1 (N)ϕ ∗ϕ ∗✲ Ω k (M)d❄✲ Ω k+1 (M)(8) L X = i X ◦ d + d ◦ i X for any X ∈ X k (M) (the Cartan ‘magic’ formula).(9) L X ◦ d = d ◦ L X , i.e., [L X , d] = 0 for any X ∈ X k (M).(10) [L X , i Y ] = i [x,y] ; in particular, i X ◦L X = L X ◦i X for all X, Y ∈ X k (M).Given a k−form α = f I dx I ∈ Ω k (M), the exterior derivative is definedin local coordinates ( x 1 , ..., x n) of a point m ∈ M asdα = d ( f I dx I) = ∂f I∂x i dx i k∧ dx I = df I ∧ dx i1 ∧ ... ∧ dx i k.kIn particular, the exterior derivative of a function f ∈ C k (M, R) is a1−form df ∈ Ω 1 (M), with the property that for any m ∈ M, and X ∈X k (M),df m (X) = X(f),i.e., df m (X) is a Lie derivative of f at m in the direction of X. Therefore,in local coordinates ( x 1 , ..., x n) of a point m ∈ M we havedf = ∂f∂x i dxi .For any two functions f, g ∈ C k (M, R), exterior derivative obeys theLeibniz rule:d(fg) = g df + f dg,