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27 Jun 2024: ϕ ) I′ tan f and Rsky [10.5(I′ tan f)2] cos f. ... sistency (e.g., by considering special cases such as ϕd = ). You may use without proofthe spherical trigonometric identity cos A = cos a sin B sin C cos B cos C, where

3 Jul 2024: Show that. ψ(x) =. {Aexp(η|x|) for |x| > a,B cos(kx) for |x| < a,. ... iii) Find A(t) in terms of cos(ϕαt). You need not calculate its normalisation explicitly.

27 Jun 2024: Findac, the corresponding critical value of a. (b) Restoring the g term, evaluate F for a smectic state ϕ(r) = A cos q0z with z anarbitrary axis. ... You may use without proof that. 2π0 | cos(u)|3 du = 8/3.

3 Jul 2024: Given boundary conditionsu(a,θ) = 0 and u(b,θ) = cos 2θ, find u explicitly. ... b) If ⟨α(x),x⟩ = 0 for all x V , then α = 0.

3 Jul 2024: J = J0. . sin(λz)cos(λz). 0. . with λ and J0 constant. ... Given boundary conditionsu(a,θ) = 0 and u(b,θ) = cos 2θ, find u explicitly.

3 Jul 2024: T̂ =. (a b. b a. ),. where a(t) and b(t) are functions of time and a, b denote the complexconjugates, such that. ( ... Assume that w(x) = cos(πx). Show that in this case, the eigenvalues of B havenon-positive real part.

3 Jul 2024: b) Consider spatial perturbations of the form cos(kx) about the solution found inpart (a), and set λ = D1/D2. ... Ai(t) π1/2 (t)1/4 cos[23(t)3/2 14π. ]as t. ]. (c) We seek approximations to the smallest eigenvalues λ, which are close to 1.Using the

4 Jul 2024: Ai(t) π1/2 (t)1/4 cos[23(t)3/2 14π. ]as t. ]. (c) We seek approximations to the smallest eigenvalues λ, which are close to 1.Using the ... S. A S. a b A A. a b b b a b.