Ivancevic_Applied-Diff-Geom

58 Applied Differential Geometry: A Modern Introductionthe following way.a ij x i x j = 1 2 a ijx i x j + 1 2 a ijx i x j .If we swap indices in the second term, we get= 1 2 a ijx i x j + 1 2 a jix j x i , which is equal to= 1 2 (a ij + a ji ) x i x j .If we now use a substitution,12 (a ij + a ji ) ≡ b ij = b ji , we geta ij x i x j = b ij x i x j ,where a ij is non–symmetric and b ij is symmetric, as required.(iii) Every second–order tensor that is the sum a ij = u i v j + u j v i , or,a ij = u i v j + u j v i is symmetric. In both cases, if we swap the indices i andj, we get a ji = u j v i + u i v j , (resp. a ji = u j v i + u i v j ), which implies thatthe tensor a ij (resp. a ij ) is symmetric.(iv) Every second–order tensor that is the difference b ij = u i v j − u j v i ,or, b ij = u i v j − u j v i is skew–symmetric. In both cases, if we swap theindices i and j, we get b ji = −(u j v i − u i v j ), (resp. b ji = −(u j v i − u i v j )),which implies that the tensor b ij (resp. b ij ) is skew–symmetric.2.1.2 Euclidean Tensors2.1.2.1 Basis Vectors and the Metric Tensor in R nThe natural Cartesian coordinate basis in an nD Euclidean space R n isdefined as a set of nD unit vectors e i given bye 1 = [{1, 0, 0, ...} t , e 2 = {0, 1, 0, ...} t , e 3 = {0, 0, 1, ...} t , ..., e n = {0, 0, ..., 1} t ],(where index t denotes transpose) while its dual basis e i is given by:e 1 = [{1, 0, 0, ...}, e 2 = {0, 1, 0, ...}, e 3 = {0, 0, 1, ...}, ..., e n = {0, 0, ..., 1}],