Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 169or, if the following diagram commutes:T I✻1I;T u✲ T M✻˙γ X ✒✲γ MOn a chart (U, φ) with coordinates φ(m) = ( x 1 (m), ..., x n (m) ) , forwhich ϕ ◦ γ : t ↦→ γ i (t) and T ϕ ◦ X ◦ ϕ −1 : x i ↦→ ( x i , X i (m) ) , this iswritten˙γ i (t) = X i (γ (t)) , for all t ∈ I ⊆ R, (3.36)which is an ordinary differential equation of first–order in n dimensions.The velocity ˙γ of the parameterized curve γ (t) is a vector–field along γdefined by˙γ(t) = (γ(t), ẋ 1 (t), . . . ẋ n (t)).Its length | ˙γ| : I → R, defined by | ˙γ|(t) = | ˙γ(t)| for all t ∈ I, is a functionalong α. | ˙γ| is called speed of γ [Arnold (1989)].Each vector–field X alongγ is of the form X(t) = (γ(t), X 1 (t), . . . , X n (t)), where each componentX i is a function along γ. X is smooth if each X i : I → M is smooth. Thederivative of a smooth vector–field X along a curve γ(t) is the vector–fieldẊ along γ defined byẊ(t) = (γ(t), Ẋ 1 (t), . . . Ẋ n (t)).Ẋ(t) measures the rate of change of the vector part (X 1 (t), . . . X n (t)) ofX(t) along γ. Thus, the acceleration ¨γ(t) of a parameterized curve γ(t) isthe vector–field along γ get by differentiating the velocity field ˙γ(t).Differentiation of vector–fields along parameterized curves has the followingproperties. For X and Y smooth vector–fields on M along theparameterized curve γ : I → M and f a smooth function along γ, we have:(1) d dt (X + Y ) = Ẋ + Ẏ ;(2) d dt (fX) = fX ˙ + fẊ; and(X · Y ) = ẊY + XẎ .(3) d dt