Ivancevic_Applied-Diff-Geom

Technical Preliminaries: Tensors, Actions and Functors 53byy 1 = x 1 cos x 2 cos x 3 , y 2 = x 1 sin x 2 cos x 3 , y 3 = x 1 sin x 3 , (2.3)with the Jacobian matrix given by⎛( ) ∂yk cos x 2 cos x 3 −x 1 sin x 2 cos x 3 −x 1 cos x 2 sin x 3 ⎞∂x i = ⎝ sin x 2 cos x 3 x 1 cos x 2 cos x 3 −x 1 sin x 2 sin x 3 ⎠ (2.4)sin x 3 0 x 1 cos x 3 ∣ ∣∣and the corresponding Jacobian determinant, ∣ ∂yk∂x = (x 1 ) 2 cos x 3 .iAn inverse transform is given byx 1 = √ ( ) y(y 1 ) 2 + (y 2 ) 2 + (y 3 ) 2 , x 2 2= arctany 1 ,()∣x 3 y 3= arctan √ , with∂x i ∣∣∣ 1(y1 ) 2 + (y 2 ) 2 ∣∂y k =(x 1 ) 2 cos x 3 .As an important engineering (robotic) example, we have a rectangulartransformation from 6 DOF external, end–effector (e.g., hand) coordinates,into n DOF internal, joint–angle coordinates. In most cases this is a redundantmanipulator system, with infinite number of possible joint trajectories.2.1.1.2 Scalar InvariantsA scalar invariant (or, a zeroth order tensor) with respect to the transformation(2.2) is the quantity ϕ = ϕ(t) defined asϕ(x i ) = ¯ϕ(¯x i ),which does not change at all under the coordinate transformation. In otherwords, ϕ is invariant under (2.2). For example, biodynamic examples ofscalar invariants include various energies (kinetic, potential, biochemical,mental) with the corresponding kinds of work, as well as related thermodynamicquantities (free energy, temperature, entropy, etc.).2.1.1.3 Vectors and CovectorsAny geometrical object v i = v i (t) that under the coordinate transformation(2.2) transforms as¯v i = v j ∂¯xi∂x j ,(remember, summing upon j−index),