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Vectors_Tensors_18_Curvilinear_Calculus

# Vectors_Tensors_18_Curvilinear_Calculus - Section 1.18...

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Unformatted text preview: Section 1.18 Solid Mechanics Part III Kelly 156 1.18 Curvilinear Coordinates: Tensor Calculus 1.18.1 Differentiation of the Base Vectors Differentiation in curvilinear coordinates is more involved than that in Cartesian coordinates because the base vectors are no longer constant and their derivatives need to be taken into account, for example the partial derivative of a vector with respect to the Cartesian coordinates is i j i j x v x e v ∂ ∂ = ∂ ∂ but 1 j i i i j i j v v Θ ∂ ∂ + Θ ∂ ∂ = Θ ∂ ∂ g g v The Christoffel Symbols of the Second Kind First, from Eqn. 1.16.3 – and using the inverse relation, k m k j i m m i m j j i x x x g e g ∂ Θ ∂ Θ Θ ∂ ∂ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Θ ∂ ∂ Θ ∂ ∂ = Θ ∂ ∂ 2 (1.18.1) this can be written as k k ij j i g g Γ = Θ ∂ ∂ Partial Derivatives of Covariant Base Vectors (1.18.2) where m k j i m k ij x x ∂ Θ ∂ Θ Θ ∂ ∂ = Γ 2 , (1.18.3) and k ij Γ is called the Christoffel symbol of the second kind ; it can be seen to be equivalent to the k th contravariant component of the vector j i Θ ∂ ∂ / g . One then has { ▲ Problem 1} k i j k j i k ji k ij g g g g ⋅ Θ ∂ ∂ = ⋅ Θ ∂ ∂ = Γ = Γ Christoffel Symbols of the 2 nd kind (1.18.4) and the symmetry in the indices i and j is evident 2 . Looking now at the derivatives of the contravariant base vectors i g : differentiating the relation k i k i δ = ⋅ g g leads to k ij k m m ij k j i i j k Γ = ⋅ Γ = ⋅ Θ ∂ ∂ = ⋅ Θ ∂ ∂ − g g g g g g 1 of course, one could express the i g in terms of the i e , and use only the first of these expressions 2 note that, in non-Euclidean space, this symmetry in the indices is not necessarily valid Section 1.18 Solid Mechanics Part III Kelly 157 and so k i jk j i g g Γ − = Θ ∂ ∂ Partial Derivatives of Contravariant Base Vectors (1.18.5) Transformation formulae for the Christoffel Symbols The Christoffel symbols are not the components of a (third order) tensor. This follows from the fact that these components do not transform according to the tensor transformation rules given in §1.17. In fact, s k j i s r pq r k j q i p k ij Θ ∂ Θ ∂ Θ Θ ∂ Θ ∂ + Γ Θ ∂ Θ ∂ Θ ∂ Θ ∂ Θ ∂ Θ ∂ = Γ 2 The Christoffel Symbols of the First Kind The Christoffel symbols of the second kind relate derivatives of covariant (contravariant) base vectors to the covariant (contravariant) base vectors. A second set of symbols can be introduced relating the base vectors to the derivatives of the reciprocal base vectors, called the Christoffel symbols of the first kind : k i j k j i jik ijk g g g g ⋅ Θ ∂ ∂ = ⋅ Θ ∂ ∂ = Γ = Γ Christoffel Symbols of the 1 st kind (1.18.6) so that the partial derivatives of the covariant base vectors can be written in the alternative form k ijk j i g g Γ = Θ ∂ ∂ , (1.18.7) and it also follows from Eqn. 1.18.2 that mk ijm k ij mk m ij ijk g g Γ = Γ Γ = Γ , (1.18.8) showing that the index...
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Vectors_Tensors_18_Curvilinear_Calculus - Section 1.18...

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