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.