Vectors_Tensors_17_Curvilinear_Transform

Vectors_Tensors_17_Curvilinear_Transform - Section 1.17...

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Section 1.17 Solid Mechanics Part III Kelly 152 1.17 Curvilinear Coordinates: Transformation Laws 1.17.1 Coordinate Transformation Rules Suppose that one has a second set of curvilinear coordinates ) , , ( 3 2 1 Θ Θ Θ , with ) , , ( ), , , ( 3 2 1 3 2 1 Θ Θ Θ Θ = Θ Θ Θ Θ Θ = Θ i i i i (1.17.1) By the chain rule, the covariant base vectors in the second coordinate system are given by j i j j i j i i g x x g Θ Θ = Θ Θ Θ = Θ = A similar calculation can be carried out for the inverse relation and for the contravariant base vectors, giving j j i i j j i i j i j i j i j i g g g g g g g g Θ Θ = Θ Θ = Θ Θ = Θ Θ = , , (1.17.2) The coordinate transformation formulae for vectors u can be obtained from i i i i u u g g u = = and i i i i u u g g u = = : j i j i j i j i j j i i j j i i u u u u u u u u Θ Θ = Θ Θ = Θ Θ = Θ Θ = , , Vector Transformation Rule (1.17.3) These transformation laws have a simple structure and pattern – the subscripts/superscripts on the transformed coordinates Θ quantities match those on the transformed quantities,
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Vectors_Tensors_17_Curvilinear_Transform - Section 1.17...

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