Ivancevic_Applied-Diff-Geom

52 Applied Differential Geometry: A Modern Introductionas expected from the original matrix form Ax = b. This indicial notationcan be more useful than the matrix one, like e.g., in computer science,where indices would represent loop variables. However, the full potentialof tensor analysis is to deal with nonlinear multivariate systems, whichare untractable by linear matrix algebra and analysis. The core of thisnonlinear multivariate analysis is general functional transformation.2.1.1.1 Transformation of CoordinatesSuppose that we have two sets of curvilinear coordinates that are single–valued, continuous and smooth functions of time, x j = x j (t), (j = 1, ..., m)and ¯x i = ¯x i (t), (i = 1, ..., n), respectively, representing trajectories of motionof some physical or engineering system. Then a general (m × n)Dtransformation (i.e., a nonlinear map) x j ↦→ ¯x i is defined by the set oftransformation equations¯x i = ¯x i (x j ), (i = 1, ..., n; j = 1, ..., m). (2.2)In case of the square transformation, m = n, we can freely exchange theindices, like e.g., in general relativity theory. On the other hand, in thegeneral case of rectangular transformation, m ≠ n, like e.g., in robotics,and we need to take care of these ‘free’ indices.Now, if the Jacobian determinant of this coordinate transformation isdifferent from zero,∣ ∂¯x i ∣∣∣∣∂x j ≠ 0,then the transformation (2.2) is reversible and the inverse transformation,x j = x j (¯x i ),exists as well. Finding the inverse transformation is the problem of matrixinverse: in case of the square matrix it is well defined, although the inversemight not exist if the matrix is singular. However, in case of the squarematrix, its proper inverse does not exist, and the only tool that we areleft with is the so–called Moore–Penrose pseudoinverse, which gives anoptimal solution (in the least–squares sense) of an overdetermined system ofequations. Every (overdetermined) rectangular coordinate transformationinduces a redundant system.For example, in Euclidean 3D space R 3 , transformation from Cartesiancoordinates y k = {x, y, z} into spherical coordinates x i = {ρ, θ, ϕ} is given