Geometrical Theory of Dynamical Systems and Fluid Flows by T. Kameb

By T. Kameb

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Adjoint transformation Adg Y , where Y (d/ds)(g ◦ ηs ◦ g −1 )|s=0 . = dηs /ds|s=0 and Adg Y = is the Lie algebra and g is an element of the group G. The operator gY g −1 may be better written as the push-forward notation, g∗ Y g −1 . 19 Most textbooks in mathematics adopt this definition. 64)) in Chapters 4 and 9, characterized with the left-invariant metric. 59), results in different signs of the Lie bracket of right- and left-invariant fields (see [AzIz95]). 18 g Manifolds, Flows, Lie Groups and Lie Algebras 29 by the representation, Y → adX Y = [X, Y ].

Length, angle and area over Σ2 , (∆s)2 = g11 (∆u1 )2 + 2g12 ∆u1 ∆u2 + g22 (∆u2 )2 . So the intersecting angle θ of two coordinate curves u1 = const and u2 = const is given by cos θ = g12 x1 , x2 ∆1 u1 ∆2 u2 =√ √ . 16) Hence, it is necessary for orthogonality of two coordinate curves on a surface to be g12 = 0 at each point. It is also the sufficient condition. Consider a small parallelogram spanned by two infinitesimal lineelements (∆1 u1 , 0) and (0, ∆2 u2 ). 1 The metric properties of a surface, such as the length, angle and area, can be expressed completely by means of the first fundamental form of the surface.

The Lagrange derivative is understood as denoting the time derivative, with respect to the fluid particle ξt x moving with the flow, of the function f (x, t). (b) Derivatives of a vector field Y (x) Now, suppose that we are given a second vector field Y (x) = Y i ∂i , and consider its time derivative along the X-flow generated by X(x). To that end, let us denote the second Y -flow generated by Y (x) as ηs with η0 = e = id. The first flow ξt transports the vector Y (x) in front of a fisherman sitting at a point x.

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