Ivancevic_Applied-Diff-Geom

Technical Preliminaries: Tensors, Actions and Functors 63Divergence. The divergence div(v i ) of a vector–field v i = v i (x i , t) isdefined by contraction of its covariant derivative with respect to the coordinatesx i = x i (t), i.e., the contraction of v i ;k , namelydiv(v i ) = v i ;i = 1 √ g∂ x i( √ gv i ).Curl. The curl curl(θ i ) of a one–form θ i = θ i (x i , t) is a second–ordercovariant tensor defined ascurl(θ i ) = θ i;k − θ k;i = ∂ x kθ i − ∂ x iθ k .Laplacian. The Laplacian ∆ϕ of a scalar invariant ϕ = ϕ(x i , t) is thedivergence of grad(ϕ), or∆ϕ = ∇ 2 ϕ = div(grad(ϕ)) = div(ϕ ;i ) = 1 √ g∂ x i( √ gg ik ∂ x kϕ).2.1.3.5 Absolute DerivativeThe absolute derivative (or intrinsic, or Bianchi’s derivative) of a contravariantvector v i along a curve x k = x k (t) is denoted by ˙¯v i ≡ Dv i /dtand defined as the inner product of the covariant derivative of v i andẋ k ≡ dx k /dt, i.e., v ;kẋk i , and is given by˙¯v i = ˙v i + Γ i jkv j ẋ k .Similarly, the absolute derivative ˙¯v i of a covariant vector v i is defined as˙¯v i = ˙v i − Γ j ik v jẋ k .Generalization for the higher order tensors is straightforward; e.g., the absolutederivative ˙¯t j kl of the third order tensor tj klis given by˙¯t j kl = ṫj kl + Γj qst s klẋ q − Γ s kqt j slẋq − Γ s lqt j ksẋq .The absolute derivative is the most important differential operator inphysics and engineering, as it is the basis for the covariant form of bothLagrangian and Hamiltonian equations of motion of many physical andengineering systems.