Ivancevic_Applied-Diff-Geom

56 Applied Differential Geometry: A Modern Introductionand the inverse metric tensor⎛1 0 0(g ij ⎜ 1) = ⎝ 0(x 1 ) 2 cos 2 x0 3 10 0(x 1 ) 2 ⎞⎟⎠ =⎛1 0 0⎜ 1⎝ 0ρ 2 cos 2 ϕ 010 0⎟⎠ . (2.7)ρ 2 ⎞Given a tensor, we can derive other tensors by raising and loweringits indices, by their multiplication with covariant and contravariant metrictensors. In this way, the so–called associated tensors to the given tensorare be formed. For example, v i and v i are associated tensors, related byv i = g ik v k and v i = g ik v k .Given two vectors, u ≡ u iproduct is given byand v ≡ v i , their inner (dot, or scalar)u · v ≡ g ij u i v j ,while their vector (cross) product (in 3D) is given by2.1.1.5 Higher–Order Tensorsu × v ≡ ε ijk u j v k .As a generalization of above tensors, consider a geometrical object Rkps i =Rkps i (t) that under the coordinate transformation (2.2) transforms as¯R kps i = R j ∂¯x i ∂x l ∂x q ∂x tlqt∂x j ∂¯x k ∂¯x p , (all indices = 1, ..., n). (2.8)∂¯xsClearly, R i kjl = Ri kjl(x, t) is a fourth order tensor, once contravariant andthree times covariant, representing the central tensor in Riemannian geometry,called the Riemann curvature tensor. As all physical and engineeringconfiguration spaces are Riemannian manifolds, they are all characterizedby curvature tensors. In case R i kjl= 0, the corresponding Riemannianmanifold reduces to the Euclidean space of the same dimension, in whichg ik = δ i k.If one contravariant and one covariant index of a tensor a set equal, theresulting sum is a tensor of rank two less than that of the original tensor.This process is called tensor contraction.If to each point of a region in an nD space there corresponds a definitetensor, we say that a tensor–field has been defined. In particular, this is avector–field or a scalar–field according as the tensor is of rank one or zero.It should be noted that a tensor or tensor–field is not just the set of its