Ivancevic_Applied-Diff-Geom

62 Applied Differential Geometry: A Modern Introductionperform another time differentiation, we getd 2 x idt 2= ∂xi d 2 y k∂y k dt 2 + ∂2 x i dy k dy m∂y k ∂y m dt dt ,which means that d2 x id 2 x idt 2dt 2is not a proper vector.is an acceleration vector only in a special case when x i are an-∂ 2 x i∂y k ∂y mother Cartesian coordinates; then = 0, and therefore the originalcoordinate transformation is linear, x i = a i k yk + b i (where a i k and bi areconstant).Therefore, d2 x idtrepresents an acceleration vector only in terms of Newtonianmechanics in a Euclidean space R n , while it is not a proper acceleration2vector in terms of Lagrangian or Hamiltonian mechanics in general curvilinearcoordinates on a smooth manifold M n . And we know that Newtonianmechanics in R n is sufficient only for fairly simple mechanical systems.The above is true for any tensors. So we need to find another derivativeoperator to be able to preserve their tensor character. The solution to thisproblem is called the covariant derivative.The covariant derivative v;k i of a contravariant vector vi is defined asv i ;k = ∂ x kv i + Γ i jkv j .Similarly, the covariant derivative v i;k of a covariant vector v i is defined asv i;k = ∂ x kv i − Γ j ik v j.Generalization for the higher order tensors is straightforward; e.g., the covariantderivative t j kl;q of the third order tensor tj klis given byt j kl;q = ∂ x qtj kl + Γj qst s kl − Γ s kqt j sl − Γs lqt j ks .The covariant derivative is the most important tensor operator in generalrelativity (its zero defines parallel transport) as well as the basis fordefining other differential operators in mechanics and physics.2.1.3.4 Covariant Form of Differential OperatorsHere we give the covariant form of classical vector differential operators:gradient, divergence, curl and Laplacian.Gradient. If ϕ = ϕ(x i , t) is a scalar field, the gradient one–form grad(ϕ)is defined bygrad(ϕ) = ∇ϕ = ϕ ;i = ∂ x iϕ.