Ivancevic_Applied-Diff-Geom

60 Applied Differential Geometry: A Modern Introductionwhere T = |T| is its norm. To get its components in curvilinear coordinates,we need first to substitute it in Cartesian basis:then to evaluate it on the slots:T ij = T mn (e m ⊗ e n )(e i , e j ),T ij = T mn e m · e i e n · e j ,and finally to calculate the other index configurations by lowering indices,by means of the metric tensor:T i j = g jm T im , T ij = g im g jn T mn .2.1.3 Covariant DifferentiationIn this subsection, we need to consider some nD Riemannian manifold M(see section (3.10.1) below) with the metric form (i.e., line element) ds 2 =g ik dx i dx k , as a configuration space for a certain physical or engineeringsystem (e.g., robotic manipulator).2.1.3.1 Christoffel’s SymbolsPartial derivatives of the metric tensor g ik (2.5) form themselves specialsymbols that do not transform as tensors (with respect to the coordinatetransformation (2.2)), but nevertheless represent important quantities intensor analysis. They are called Christoffel symbols of the first kind, definedbyΓ ijk = 1 (2 (∂ x kg ij + ∂ x j g ki − ∂ x ig jk ), remember, ∂ x i ≡ ∂ )∂x iand Christoffel symbols of the second kind, defined byΓ k ij = g kl Γ ijl .(2.8) of the manifold M, can be ex-The Riemann curvature tensor Rijk lpressed in terms of the later asR l ijk = ∂ x j Γ l ik − ∂ x kΓ l ij + Γ l rjΓ r ik − Γ l rkΓ r ij.For example, in 3D spherical coordinates, x i = {ρ, θ, ϕ}, with the metrictensor and its inverse given by (2.6, 2.7), it can be shown that the only