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In this paper, we substituted the ordinary derivative with the covariant derivative in simple translation, that is built by applying the ordinary derivative and we investigate the new display of the simple translation. We have introduced a new translation operator in which its generator is exactly covariant derivative. To know the translation of a quantum particle around a space where the effect of gravity is significant, such as a black hole, it is enough to put the metric of that space in the translation derived in this paper. [ABSTRACT FROM AUTHOR]

The generalized momentum operators are derived in the framework of non-relativistic quantum mechanics, taking into account an especial assumption on the covariant derivative. According to this assumption, a scalar density doesn't change under the action of covariant derivative. The Hermitian form of the covariant derivative is discussed. It is shown that, with the help of the adjoint of covariant derivatives, generalized momentum operators can be derived. The inverse of covariant derivative is calculated. It is shown that the inverse of generalized momentum operators can be deduced from the inverse of covariant derivatives. [ABSTRACT FROM AUTHOR]

In this paper, the problems of covariant differentiation of two-point pseudotensor fields of arbitrary ranks and weights are considered and discussed. The requisite equations and notions from algebra and analysis of pseudotensors are recalled. The notion of a two-point pseudotensor field is introduced, based on the equation for its transformation under changing coordinate net. An equation for the transformation of a two-point pseudotensor field to a two-point absolute tensor field is proposed. Definitions of total and explicit operators of partial and covariant differentiation of two-point pseudotensors are introduced. Equations for covariant derivatives of a two-point pseudotensor field are obtained. Important examples of two-point pseudotensor fields from non-linear solid mechanics are noted. [ABSTRACT FROM AUTHOR]

The paper deals with an algebraic algorithm for the pseudotensors weights eliminating and recovering. The proposed algorithm allows us to reduce pseudotensors of arbitrary ranks and weights to absolute tensors of higher ranks. The weight of a pseudotensor is assumed to be an integer (positive or negative). The algorithm is based on the transformation of a pseudotensor of arbitrary rank and integer weight by using tensor product of permutation symbols. The requisite equations from algebra and analysis of pseudotensors are given and discussed. Based on the proposed algebraic algorithm, covariant constancy of permutation symbols and fundamental orienting pseudoscalar powers, a realisation of covariant differentiation of a pseudotensor field of arbitrary rank and integer weight is developed. The definition of the pseudotensor field gradient is then introduced. [ABSTRACT FROM AUTHOR]

Commutation formulae with respect to a non-symmetric affine connection are obtained in this paper. The components of commutation formulae in this paper are covariant derivatives of tensors with respect to symmetric and non-symmetric affine connection. [ABSTRACT FROM AUTHOR]

A simple algorithm for calculating Christoffel symbols, a covariant projection of the result of the Laplace operator's action on the vector, vector curl and other similar operations in an arbitrary oblique base are proposed. For an arbitrary base with ortho ei is found the expressions of vector projections (ΔA)i and (rotA)i, where A is a counter variant vector. Examples of orthonormal bases are considered and general expressions for (ΔA)i and (rotA)i for the bases are also given. As a demonstration of the working capacity of the common formulas obtained, detailed calculations of (ΔA)i and (rotA)i as an example are made in cases of spherical and cylindrical coordinate systems.

What today we call Truesdell rate of the Cauchy stress was originally introduced by Truesdell, as a frame-indifferent measure of stress rate for his theory of hypoelasticity. This work aims at showing that the Truesdell rate is in fact a more general concept, which arises from the application of Reynolds' Transport Theorem. In the customary three-dimensional setting, the Transport Theorem can be formulated in terms of differential forms, i.e., three-forms for volume integrals and two-forms for surface integrals, in which case the only differential operator involved is the Lie derivative with respect to the velocity field of the continuum body. Alternatively, the Transport Theorem can be formulated in terms of the pseudo-scalar density associated with a three-form and the pseudo-vector flux density associated with a two-form. In this alternative approach, the differential operator involved is precisely the Truesdell rate. As an example, we shall show how the "convected derivative" of the theory of Electromagnetism is in fact a Truesdell rate. [ABSTRACT FROM AUTHOR]

This paper shows how gauge theoretic structures arise in a noncommutative calculus where the derivations are generated by commutators. These patterns include Hamilton's equations, the structure of the Levi–Civita connection, and generalizations of electromagnetism that are related to gauge theory and with the early work of Hermann Weyl. The territory here explored is self-contained mathematically. It is elementary, algebraic, and subject to possible generalizations that are discussed in the body of the paper. [ABSTRACT FROM AUTHOR]

We present detailed pedagogical derivation of covariant derivative of fermions and some related expressions, including commutator of covariant derivatives and energy-momentum tensor of a free Dirac field. On top of that, local conformal transformations for a Dirac fermion in curved spacetime are considered and we obtain the expression for the energy-momentum tensor on the cosmological background.

In this paper, we have studied the second covariant derivative of Riemannian curvature tensor. Some new identities for the second covariant derivative have been given. Namely, identities obtained by cyclic sum with respect to three indices have been given. In the first case, two curvature tensor indices and one covariant derivative index participate in the cyclic sum, while in the second case one curvature tensor index and two covariant derivative indices participate in the cyclic sum. [ABSTRACT FROM AUTHOR]

In [22], V.M. Stankovíc introduced generalized Einstein type tensors of the kind θ (θ = 1, 2) in the generalized Riemannian space, and some relations which they satisfy are obtained using the first and the second kind of covariant derivative. In the present paper, generalized Einstein type tensors of the kind θ (θ = 3, 4) are introduced. Also, some relations which Einstein type tensors of the kind θ (θ = 1, 2, 3, 4) were found using all four kinds of covariant derivative. [ABSTRACT FROM AUTHOR]

This paper shows how gauge theoretic structures arise in a noncommutative calculus where the derivations are generated by commutators. These patterns include Hamilton’s equations, the structure of the Levi–Civita connection, and generalizations of electromagnetism that are related to gauge theory and with the early work of Hermann Weyl. The territory here explored is self-contained mathematically. It is elementary, algebraic, and subject to possible generalizations that are discussed in the body of the paper.

In this manuscript, the identities of Ricci Type with respect to a non-symmetric affine connection space are obtained and simplified. The components of commutation formulae are discussed. [ABSTRACT FROM AUTHOR]

We present a further development of the algebraic aspect of gauge theories initiated in our previous papers. [ABSTRACT FROM AUTHOR]

Let M be a real hypersurfaces in a complex space form Mn(c), c ≠ 0, whose Lie derivative of shape operator in the direction of the Reeb vector field coincides with the covariant derivative of it in the same direction. In this paper, we characterize such real hypersurfaces of Mn(c). [ABSTRACT FROM AUTHOR]

Let H be a vector-valued holomorphic reproducing kernel function space and H 0 n be the subspace of functions vanishing to order n on an analytic hyper-surface. We determine unitary equivalence of the quotient space H ⊖ H 0 n by covariant derivatives of the curvature on the vector bundle associated to the reproducing kernel of H. The proof is a combination of some geometric and combinatorial results. In particular, we show that on a holomorphic Hermitian vector bundle, matrix representations for covariant derivatives of its curvature satisfy a recursive formula, which is of independent interest. [ABSTRACT FROM AUTHOR]