Jacobi–Lie Bracket

	Jacobi–Lie Bracket In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields X and Y on a smooth manifold M a third vector field denoted ,Y/math>. Conceptually, the Lie bracket ,Y/math> is the derivative of Y along the flow generated by X, and is sometimes denoted ''\mathcal_X Y'' ("Lie derivative of Y along X"). This generalizes to the Lie derivative of any tensor field along the flow generated by X. The Lie bracket is an R- bilinear operation and turns the set of all smooth vector fields on the manifold M into an (infinite-dimensional) Lie algebra. The Lie bracket plays an important role in differential geometry and differential topology, for instance in the Frobenius integrability theorem, and is also fundamental in the geometric theory of nonlinear control systems. V. I. Arnold refers to this as the "fisherman derivative", as one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Differential Topology In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which concerns the ''geometric'' properties of smooth manifolds, including notions of size, distance, and rigid shape. By comparison differential topology is concerned with coarser properties, such as the number of holes in a manifold, its homotopy type, or the structure of its diffeomorphism group. Because many of these coarser properties may be captured algebraically, differential topology has strong links to algebraic topology. The central goal of the field of differential topology is the classification of all smooth manifolds up to diffeomorphism. Since dimension is an invariant of smooth manifolds up to diffeomorphism type, this classification is often studied by classifying the ( connected) manifolds in each dimension separately: * In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Directional Derivative In multivariable calculus, the directional derivative measures the rate at which a function changes in a particular direction at a given point. The directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a direction specified by v. The directional derivative of a scalar function ''f'' with respect to a vector v at a point (e.g., position) x may be denoted by any of the following: \begin \nabla_(\mathbf) &=f'_\mathbf(\mathbf)\\ &=D_\mathbff(\mathbf)\\ &=Df(\mathbf)(\mathbf)\\ &=\partial_\mathbff(\mathbf)\\ &=\mathbf\cdot\\ &=\mathbf\cdot \frac.\\ \end It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant. The directional derivative is a special case of the Gateaux derivative. Definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Left Invariant Left may refer to: Music * ''Left'' (Hope of the States album), 2006 * ''Left'' (Monkey House album), 2016 * ''Left'' (Helmet album), 2023 * "Left", a song by Nickelback from the album ''Curb'', 1996 Direction * Left (direction), the relative direction opposite of right * Left-handedness Politics * Left (Austria), a movement of Marxist–Leninist, Maoist and Trotskyist organisations in Austria * Left-wing politics (also known as left or leftism), a political trend or ideology Centre-left politics Far-left politics * The Left (Germany) See also * Copyleft * Leaving (other) * Lefty (other) * Sinister (other) * Venstre (other) * Right (other) A right is a legal or moral entitlement or permission. Right or rights may also refer to: * Right, synonym of true or accurate, opposite of wrong * Morally right, opposite of morally wrong * Right (direction), the relative direction opposite of ... {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Lie Group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (to allow division), or equivalently, the concept of addition and subtraction. Combining these two ideas, one obtains a continuous group where multiplying points and their inverses is continuous. If the multiplication and taking of inverses are smoothness, smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the circle group. Rotating a circle is an example of a continuous symmetry. For an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Frobenius Integrability Theorem In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations. In modern differential geometry, geometric terms, given a family of vector fields, the theorem gives necessary and sufficient integrability conditions for the existence of a foliation by maximal integral manifolds whose tangent bundles are spanned by the given vector fields. The theorem generalizes the Picard–Lindelöf theorem, existence theorem for ordinary differential equations, which guarantees that a single vector field always gives rise to integral curves; Frobenius gives compatibility conditions under which the integral curves of ''r'' vector fields mesh into coordinate grids on ''r''-dimensional integral manifolds. The theorem is foundational in differential topology and Differentiable manifold, calculus on manifolds. Contact geometry studies 1-forms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]