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2.8 The Lie derivative 21Theorem 2.5 For the exterior derivative dα of a p-form α we haved(Φ ∗ α)=Φ ∗ (dα). (2.51)Proof: Let us denote local coordinates in corresponding neighbourhoodsof p ∈Mand Φ(p) ∈N by (x 1 ,...,x m ) and (y 1 ,...,y n ) respectively.Obviously, (2.51) is true for a 0-form f:d(Φ ∗ f)= ∂(Φ∗ f)∂x kdx k = ∂f(y(x))∂y i∂y i∂x k dxk =Φ ∗ (df). (2.52)Suppose the relation is valid for the (p − 1)-form β and let α = fdβ.(This is sufficiently general.) Then,d(Φ ∗ α) = d[(Φ ∗ f)d(Φ ∗ β)] = d(Φ ∗ f) ∧ d(Φ ∗ β)=Φ ∗ (dα). (2.53)We do not consider integration on manifolds, except to note that theoperator d of exterior derivation has been defined so that Stokes’s theoremcan be written in the simple form∫ ∫α = dα, (2.54)∂Vwhere α isany(k −1)-form and ∂V denotes the oriented boundary of a k-dimensional manifold with boundary V. (An n-dimensional manifold withboundary is defined by charts which map their neighbourhoods U into thehalf space H n defined by x n ≥ 0 rather than into E n , the boundary thenbeing the set of points mapped to x n = 0.)V2.8 The Lie derivativeFor each point p ∈M, a vector field v on M determines a unique curveγ p (t) such that γ p (0) = p and v is the tangent vector to the curve. Thefamily of these curves is called the congruence associated with the vectorfield. Along a curve γ p (t) the local coordinates (y 1 ,...,y n ) are thesolutions of the system of ordinary differential equationsdy idt = vi (y 1 (t),...,y n (t)) (2.55)with the initial values y i (0) = x i (p).To introduce a new type of differentiation we consider the map Φ t draggingeach point p, with coordinates x i , along the curve γ p (t) through p intothe image point q =Φ t (p) with coordinates y i (t). For sufficiently smallvalues of the parameter t the map Φ t is a one-to-one map which induces