By Joseph A. Goguen (auth.), Hélène Kirchner, Wolfgang Wechler (eds.)
This quantity comprises papers provided on the moment overseas convention on Algebraic and good judgment Programming in Nancy, France, October 1-3, 1990.
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Additional resources for Algebraic and Logic Programming: Second International Conference Nancy, France, October 1–3, 1990 Proceedings
23. 37. 126). 38. Verify that A−1 k = Ak −1 = A−k , where k = 1, 2, 3, . . 39. 134). 40. 3). 41. Verify that [(a ⊗ b) (c ⊗ d)] : I = (a · d) (b · c). 42. Express trA in terms of the components Ai·j , Aij , Aij . 43. Prove that M : W = 0, where M is a symmetric tensor and W a skewsymmetric tensor. 44. Evaluate trWk , where W is a skew-symmetric tensor and k = 1, 3, 5, . . 45. Verify that sym (skewA) = skew (symA) = 0, ∀A ∈ Linn . 46. Prove that sph (devA) = dev (sphA) = 0, ∀A ∈ Linn . 1 Vector- and Tensor-Valued Functions, Diﬀerential Calculus In the following we consider a vector-valued function x (t) and a tensor-valued function A (t) of a real variable t.
N). Thus, we can write n M= Mii g i ⊗ g i + i=1 n Mij (g i ⊗ g j + g j ⊗ g i ) , M ∈ Symn . 150) i,j=1 i>j taking into account that Wii = 0 and Wij = −Wji (i = j, i, j = 1, 2, . . , n). Therefore, the basis of Symn is formed by n tensors g i ⊗ g i and 12 n (n − 1) tensors g i ⊗g j +g j ⊗g i , while the basis of Skewn consists of 12 n (n − 1) tensors g i ⊗ g j − g j ⊗ g i , where i > j = 1, 2, . . , n. Thus, the dimensions of Symn and Skewn are 12 n (n + 1) and 12 n (n − 1), respectively.
3. Evaluate gradients of the following functions of r: (a) 1 , (b) r · w, (c) rAr, (d) Ar, (e) w × r, r where w and A are some vector and tensor, respectively. 4. 144). 5. 87). 6. 94). 7. 21. 8. 144). 9. Evaluate tangent vectors, metric coeﬃcients and Christoﬀel symbols for cylindrical surface coordinates deﬁned by s s r (r, s, z) = r cos e1 + r sin e2 + ze3 . 10. 145). 11. 130). 12. 144), respectively. 13. Prove that the Laplacian of a vector-valued function t (r) can be given by Δt = t,i|i . Specify this identity for Cartesian coordinates.
Algebraic and Logic Programming: Second International Conference Nancy, France, October 1–3, 1990 Proceedings by Joseph A. Goguen (auth.), Hélène Kirchner, Wolfgang Wechler (eds.)