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Unformatted text preview: A.3. CURVILINEAR COORDINATES 75 A.3.1 Tensor invariance Let’s presume that x i is the old Cartesian coordinate system, and ˜ x j represents the new curvilinear coordinate system. Both systems are related by transforma- tion rules (A.4) and (A.10). The distance between the material points should be invariant , i.e. independent on the coordinate system, thus: (A.34) dl 2 dx i dx i b i j d ˜ x j b i k d ˜ x k b i j b i k d ˜ x j d ˜ x k Definition A.3.1 Metric Tensor The metric tensor , g ij is defined from the relation (A.34) as: (A.35) g jk b i j b i k ∂ x i ∂ ˜ x j ∂ x i ∂ ˜ x k The metric tensor is also called the fundamental tensor. Using (A.35), we can rewrite (A.34) as: (A.36) dl 2 g ij d ˜ x i d ˜ x j The inverse of the metric tensor is also called the conjugate metric tensor, g ij , which satisfies the relation: (A.37) g ik g kj δ ij (see Problem 2.7.7). Definition A.3.2 Conjugate tensors For each index of a tensor we introduce the conjugate tensor where this index is transfered to its counterpart (covariant/contravariant) using the relations: A i g ij A j A i g ij A j (A.38) 76 APPENDIX A. INTRODUCTION TO TENSOR CALCULUS Conjugate tensor is also called the associate tensor . Relations (A.38) are also called as operations of raising/lowering of indexes (Problem A.4.6)....
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- Spring '11
- Tensor, Covariance and contravariance of vectors, Metric tensor