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- Determine the Miller indices for the directions A, B, C and D shown in the following cubic unit cell.Which of the following gives the correct solution(s) for the equation 2x = x - 4 ? O A. no solution O B. {2} O C. {8} O D. {2,8}3. Let a, b, n be integers, with n > 0. Prove that a = b (mod n) if and only if a² + b² = 2ab (mod n²).
- Elliptical cylindrical (EC) coordinates are defined by the variables u, v and z, with aconstant a, in terms of Cartesian coordinates as follows:x = a cosh u cos v ;y = a sinh u sin v ; (75)z = z .a. Find the EC metric.b. Describe surfaces of constant u, v and z for various values of the constant a.c. Find the EC basis vectors for arbitrary a.d. Make vector field plots of the EC basis vectors in the x-y plane.e. Find the incremental area and volume expressions.f. Express the gradient, divergence, curl and Laplacian operators in EC coordinates.ex12. Let u, v, w and x be vectors. Classify the following expressions as being one of the following: vector, real number, meaningless. (a) (ux v) x (w x x), (b) (ux v). (w x x), (c) (ux v) (w xx), (d) u. (v x W),