covariant vector

covariant vector

[kō′ver·ē·ənt ′vek·tər]
(mathematics)
A covariant tensor of degree 1, such as the gradient of a function.
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n] be an n-dimensional differentiable manifold endowed with a (1,1) tensor field [phi], a contravariant vector field [xi], a covariant vector field [eta] and a Lorentzian metric g of type (0, 2) such that for each point p [member of] M, the tensor [g.
Starting from generic bilinear Hamiltonians, constructed by covariant vector, bivector or tensor fields, it is possible to derive a general symplectic structure which leads to holonomic and anholonomic formulations of Hamilton equations of motion directly related to a hydrodynamic picture.
We show that a covariant analogue of Hamilton equations can be derived from covariant vector (or tensor) fields in holonomic and anholonomic coordinates.
Analogously, for the covariant derivative applied to the covariant vector,
In fact, taking into account generic Hamiltonian invariants, constructed by covariant vectors, bivectors or tensors, it is possible to show that a symplectic structure can be achieved in any case.
where the invariant can be constructed by covariant vectors, bivectors and tensors.