By Elcio Abdalla; et al

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**Example text**

Carrying out these operations gives us ∂gµν ∂xα Aµ Aν d x α + gµν Aµ δ Aν + gµν Aν δ Aµ = 0. 22) to eliminate δAµ and δAν gives us ∂gµν ∂xα − gνβ β να − gνβ β µα = 0. 36) β Now, να is symmetric in the lower indexes ν and α, and this symmetry allows permutation of ν and α to obtain ∂gµα − gµβ β να − gαβ β µν = 0. 37) ∂gνα − gνβ ∂xµ β µα − gαβ β µν = 0. 38) for γ µα = γ µα , we obtain ∂gµα ∂g 1 γ ν ∂gνµ . 39) The Christoffel symbol of the first kind is ν,µα = ∂g ∂g 1 ∂gνµ . 40) It is often written as [µα, ν].

Two tensors, Aµν and B µν , are said to be reciprocal to each other if Aµν B να = δα µ . 15) A tensor is called symmetric with respect to two contravariant or two covariant indexes if its components remain unaltered on interchange of the indexes. For exµνα νµα ample, if Aβγ = Aβγ , the tensor is symmetric in µ and ν. If a tensor is symmetric with respect to any two contravariant and any two covariant indexes, it is called symmetric. A tensor is called skew-symmetric with respect to two contravariant or two covariant indexes if its components change sign upon interchange of the indexes.

Prove that the contraction of the outer product of the two tensors A p and B p is a scalar. 7. Determine the Christoffel symbols of the second kind in rectangular and cylindrical coordinates. 8. The line element on the surface of a sphere of radius a in Euclidean space is given by ds 2 = a 2 (dθ 2 + sin2 θ dφ 2 ). For this space calculate i kl , i, k, l = 1, 2 (with θ = x1 , φ = x2 ). 9. Find the covariant derivative of Ai j B km n with respect to x q . 10. 11. Determine the force acting on a particle in a constant gravitational field.