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4 Oct 2023: S(t) = cos 2ωtS (1 cos 2ωt) B̂ (B̂ S) sin 2ωt B̂S. ... H = γB(t) S where B(t) = B0ẑ b (x̂ cos ω1t ŷ sin ω1t).

1 Nov 2023: 5. Diagonalise the quadratic form in R2 defined by. F(x, y) = (a cos2 θ b sin2 θ) x2 2(a b)(sin θ cos θ) xy (a sin2 θ b

16 Jan 2024: Find the time-averaged energy flux vector 〈 I 〉 for (i) a plane harmonic S-wave with u =. B cos(k x ωt), (ii) a plane harmonic P-wave with u = Ak̂ ... An SV-wave with displacement. u = B(cosθ,0, sinθ)eik(x sin θz cos θ)iωt.

23 Oct 2023:   cos φ sin φ 0sin φ cos φ 00 0 1. ... ψ̇ φ̇ cos θ. . (b) [optional extra] Calculate the matrix B = R>Ṙ and verify that it is antisymmetric.

6 Oct 2023: I0(x) =1. π. π. 0. ex cos θ dθ. Oscillatory Integrals and Stationary Phase. ... a). 1. 0. cos(. x tp). dt , with p > 1 , real ,. (b). π. 2. 0. (. 1 (. 2θ. π. ))γ. cos(.

2 Feb 2024: Let there be a magnetic field B, sothat each molecule has a potential energy mB cos θ (θ being the angle between theaxis of the molecule and the magnetic field). ... N where. zrot =. [. 2I. 2mBβ2. ]. sinh(mβB). Evaluate the total magnetisation, M =

15 Jan 2024: What. motivated your choice? [Suggestions: You could try ψ = cos(πx/2α) or ψ = (α2 x2) for |x| < α and vanishing. ... b. A defect is introduced into the crystal. As a result, the amplitude for jumping.

1 Nov 2023: 10. The linear map R3 R3 defined by. x 7 x′ = cos θ x (x n) (1 cos θ) n sin θ (x n) (). is a rotation by angle θ in ... Hence show that. Rii = 2 cos θ 1 , εijkRjk = 2ni sin θ.

23 Oct 2023: q1 = Q1 cos λ P2 sin λ, q2 = Q2 cos λ P1 sin λ,. ... p1 = Q2 sin λ P1 cos λ, p2 = Q1 sin λ P2 cos λ.

8 Mar 2024: dt=. 1. Ra, where R2 = k1. Derive the. equationsa(η) = B(1 cos η) , t(η) = BR(η sin η) ,. where B is a constant, and hence show that the universe ... ds2 = sin θ (dφ2 dθ2) 2 cos θ dθ dφ ,. where 0 θ 2π and 0 φ 2π. Show that, for a null

18 Oct 2012: y =. . . . . . 0 , x < 0 ,. 1 cos x , 0 < x < ǫ ,. cos(x ǫ) cos x , x > ǫ. ... 1 cos x. x. )2. dx = π. (b) Use Parseval’s theorem and the result of question 7b to evaluate the integral.

26 Jan 2024: Show that the resulting state is. |ψαβ⟩ = 12 (cos(α β) |00⟩ sin(α β) |01⟩ sin(α β) |10⟩ cos(α β) |11⟩). ... B) to be independent of the choiceof angle chosen and used remotely at B (resp.

31 Oct 2012: x = ax′ , y = by′ , z = cz′ ,. x′ = r′ sin θ′ cos φ′ , y′ = r′ sin θ′ sin φ′ , z′ = r′ cos θ′ ,. where a, b and c are positive constants. Calculate ... If an extra transverse force F sin(πx/L) cos(πct/L) per unit length

25 Jan 2007: es = cos φ i sin φ j, eφ = sin φ i cos φ j, ez = k. ... For (iii), from a decent diagram of spherical polars, find k = er cos θ eθ sin θ. ].

16 Oct 2021: Γ. θ. r. θ. =. Γ. r. θ. θ. =. 1. r. Γ. ϕ. ϕ. θ. =. -. sin. θ. cos. ... θ. ϕ. =. cos. θ. sin. θ. (1.11). There are some important lessons here.

4 Oct 2013: cos A cos B =1. 2[cos(AB) cos(A B)]. sin A sin B =1. ... 2[cos(AB) cos(A B)]. sin A cos B =1. 2[sin(A B) sin(AB)].

27 Mar 2022: We write. 𝐞. a. ′′. =. R. 1. (. θ. ). a. b. 𝐞. b. ′. with. R. 1. (. θ. ). =. (. 1. 0. 0. 0. cos. θ. sin. θ. 0. -. ... V. eff. (. θ. ). =. I. 1. (. b. -. a. cos. θ. ). 2. 2. sin. 2. θ. M. g.

1 Nov 2023: For b > 0 and x R, we have. cos(x ib) = 12(ei(xib) ei(xib)) = 1. ... 2(ebeix ebeix). and thus Re (cos(x ib)) = cosh(b) cos(x) and Im (cos(x ib)) = sinh(b) sin(x).

9 Jan 2024: given boundary conditions (λ a constant):. (i) r < R with ψ(R,θ) = λ cos θ,. ... ii) r > R with ψ(R,θ) = λ cos θ and ψ(r,θ) 0 as r ,. (iii) a < r < b with n ψ(a,θ) = 0 and ψ(b,θ) =

10 Mar 2020: ȧ. a= ḃ. b= S sin θ cos θ, θ̇ =. S(b2 cos2 θ a2 sin2 θ)a2 b2. Combine these effects to obtain. ... ȧ. a= ḃ. b= S sin θ cos θ,. θ̇ =S(b2 cos2 θ a2 sin2 θ).