By Genov G.K.

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For the sphere. 5) Verify that for the scalar ﬁeld Ψ the following relation takes place: ∆Ψ = div ( grad Ψ) . Hint. Use the fact that grad Ψ = ∇Ψ = ei ∂i Ψ is a covariant vector. Before taking div , which is deﬁned as an operation over the contravariant vector, one has to rise the index: ∇i Ψ = g ij ∇j Ψ. Also, one has to use the metricity condition ∇i gjk = 0. 6) Derive grad ( div A) for an arbitrary vector ﬁeld A in polar coordinates. 7) Prove the following relation between the contraction of the Christoﬀel symbol Γkij and the derivative of the metric determinant g = det gµν : Γjij = 1 ∂g √ = ∂i ln g .

23). 22) for the derivation of the div A in polar coordinates. 23) x = ρ cosh χ , y = ρ sinh χ . 24) and hyperbolic coordinates 51 9) Prove the relation 1 √ ik g ij Γkij = − √ ∂i gg g and use it for the derivation of ∆Ψ in polar, elliptic and hyperbolic coordinates. 1 Basic deﬁnitions and relations Here we consider some important operations over vector and scalar ﬁelds, and relations between them. In this Chapter, all the consideration will be restricted to the special case of Cartesian coordinates, but the results can be easily generalized to any other coordinates by using the tensor transformation rule.

For this end we shall use the following strategy. First of all we choose a name for this new derivative: it will be called covariant. After that the problem is solved through the following deﬁnition: Def. 1. The covariant derivative ∇i satisﬁes the following two conditions: i) Tensor transformation after acting to any tensor; ii) In a Cartesian coordinates {X a } the covariant derivative coincides with the usual partial derivative ∂ . ∇a = ∂a = ∂X a As we shall see in a moment, these two conditions ﬁx the form of the covariant derivative of a tensor in a unique way.

### A basis of identities of the algebra of third-order matrices over a finite field by Genov G.K.

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