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16 May 2019: SIGMAA. Calculate phase probabilities from |F. O. |, |F. C. |,. C. Combine MIR phase information using Hendrickson-Lattman coefficients. p(. ). exp[A cos(. ) ... B sin(. ) C cos(2. ) D sin(2. )]. Produce 4 types of map coefficients:.

30 Aug 2019: cos a = cos b cos c sin b sin c cos A. ... sin a cos B = cos b sin c sin b cos c cos A.

17 Jun 2019: cos α =(a b) (a c)|a b||a c|. Using identities (1) and (2), prove that. ... cos δ(B, C) = cos δ(A, B) cos δ(A, C) sin δ(A, B) sin δ(A, C) cos α.

23 Nov 2019: The. Hamiltonian felt by the protons will then be. H = µ (B b cos(ωt) ) = 2µ. ... The TDSE for this system is thus(ȧ. ḃ. )= i. (B b cos(ωt).

17 Jun 2019: 1]. (b). J = π/2. 0. sin x cos x dx. ... 4]. (d) The surface S of an ellipsoid is defined parametrically byx = (b sin θ cos φ, b sin θ sin φ, a cos θ), where 0 6 θ 6 π

1 May 2019: χ2i wi. λab. (Kab K0)2 (18). C00 = 1 (19). C10 = cos θ (20). ... C20 =12(3 cos. 2 θ 1) (21). 2. Table 1: Parameters of PAH anisotropic atom-atom potential in a.u.Atom Pair laκa lbκb ρ α C6C C 00 00

16 May 2019: A cos(αφ. 1. ) B cos(αφ. 2. ). This is where the vector representation of waves becomes very useful. ... i (cosα. 1. sinα. 2. cosα. 2. sinα. 1. ). = cos(α. 1. α. 2. ) i sin(α. 1. α. 2. ).

6 Nov 2019: A cos(ka) B sin(ka) = 0 (4.23). and hence. A cos(ka) = 0 and B sin(ka) = 0. ... or B = 0 , cos(ka) = 0 ,. ψ(x) = A cos(kx) , k =nπ.

8 Nov 2019:   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.

17 Jun 2019: 1]. B C. A. D. 2. 5. Solve for x in the interval 0 6 x 6 π: cos 2x = sinx. ... 1]. Natural Sciences IA, Paper 1. 5. SECTION B. 11Z. (a) Use De Moivre’s theorem to express 8 cos(6θ) 15 sin(4θ) sin(2θ) in terms of cos θand

17 Jun 2019: Prove the Pythagorean theorem for spherical geometry in the form. cos c = cos a cos b. ... E = (E0 cos(kz ωt), 0, 0) B = (0,B0 cos(kz ωt), 0),.

30 Aug 2019: 6|1〉B, sin. π. 6|0〉B cos. π. 6|1〉B. ). ,(. sinπ. 6|0〉B cos. ... π. 6|1〉B, cos. π. 6|0〉B sin. π. 6|1〉B. ). determine the condition, which α and β must satisfy in order for the measurementstatistics to violate the

16 May 2019: Can you remember how to work with the following expression? A cos(αφ. ... 1. ) B cos(αφ. 2. ). This is where the vector representation of waves becomes very useful.

17 Jun 2019: 2]. 3 Differentiate(a) ln(x3);. [1]. (b) x cos(2x). [1]. 4 Integrate. ... a) y = (ln x)2 for 0 < x < ;. [1]. (b) y = sin(x) cos(x) for π 6 x 6 π.

27 May 2019: 1, 2), B = B(r){cos(ot),sin(ot),0}is the background magnetic field, ẑ is the normal to the air–water interface, B(r) is the strength of the ... 4h2sintyðtÞ; (18). where n̂ = {sin[c(t)],cos[c(t)],0}, b 3mB0ðC

17 Jun 2019: sationσ(u, v) =. ((a b cos u) cos v, (a b cos u) sin v, b sin u. ),. ... 2π. 0. sin2 θdθ. a b cos θ=. 2π. b2. [a.

30 Aug 2019: X(θ) = b(cos θ, sin θ, θ tan φ), 0 6 θ 6 L cos φ/b, (). where φ is the pitch of the helix (i.e. the constant angle ... between dX/dθ and the xy-plane)and L cos φ b.

17 Jun 2019: π2. 8=. n=1. 1(2n 1)2. [17]. 3B Consider the matrix. A =(. cos θ sin θsin θ cos θ. ). (a) Determine the eigenvalues of A. ... r = (a cos φ sin θ, b sin φ sin θ, c cos θ).

17 Jun 2019: cos α =(a b) (a c)|a b||a c|. Using identities (1) and (2), prove that. ... cos δ(B, C) = cos δ(A, B) cos δ(A, C) sin δ(A, B) sin δ(A, C) cos α.

30 Aug 2019: χ =(A cos αr. B sin αrr. cr2). sin2 θ. Find conditions on A,B,C and α for which B = 0 on the sphere r = a, and givean expression for ... πc(τ) sin 2πz,. χ =1k. [πd(τ). ( 8p. )1/2sin kx cos πz e(τ) sin 2kx. ],.

17 Jun 2019: b)dy. dxsin x y cos x = 12 sin 2x [5]. ... 2yx=y. (2x2 y)dxdy. Evaluate the integral. [10]. (b) The equation r = (1 cos θ) defines a closed surface in three dimensions.

17 Jun 2019: σ(u,v) = ((a b cos u) cos v, (a b cos u) sin v,b sin u),. ... ψ(x) =15a. cos(πx. 2a. ). 25a. sin(πxa. ),. in the region a x a, and zero otherwise.

19 Jul 2019: cos. (B C. 2. ) sin. (A. 2. )= 2 sin. ... G1. (i) AC = 1, B = C =π. 3radians; (ii) cos C = cos A =. 3. 4, cos B = 1. 8. 12. G2. (i) 6 π; (ii)3π.

17 Jun 2019: φ(r,θ) = A B log r. n=1. [rn(An cos nθ Bn sin nθ) rn(Cn cos nθ Dn sin nθ)]. ... where An, Bn, Cn and Dn are constants. [10]. Obtain the solution for φ(r,θ) given the boundary conditions φ(a,θ) = U andφ(b,θ) = V cos θ.

17 Jun 2019: u(x,y,0) =. m=1. n=1. Amne β. 2x sin(mπx) cos. (. 2n12 πy. ). ,. ... exp(M) =. . . . . cosha sinha 0 0sinha cosh a 0 00 0 cos b sinb0 0 sinb cos b.

17 Jun 2019: Φ(r, θ) =. n=0. (Anr. n Bnrn1)Pn(cos θ). for constants An, Bn. ... The temperatureon the boundaries of the shell is. T (a, θ) = T0 T1 cos θ, T (b, θ) = T2 T3 cos θ ,. for constants T0, T1, T2, T3. Determine the

17 Jun 2019: By settingthe former equal to cos θ|a|b̂, where b̂ is a unit vector along b, show that. ... cos θ =a b|a||b|. Explain why this is a sensible definition of the angle θ between a and b.

30 Aug 2019: 1 < p <. Letf (x) = a cos nx b sin nx,. ... n. b) Let. f (x) =. k=0. ak cos 5kx, ak > 0,.

30 Aug 2019: b) By considering f (x) = cos kx show that we cannot have a small-o estimate. ... q(x)| 6 n1 x2. at the n zeros of Tn ,. then|q(x)| 6 |T ′. n(x)| , |x| > cos π. 2n.

17 Jun 2019: A = cos(. θ. 2. )I sin. (θ. 2. )E1E2,. where θ is a scalar. ... 6]. (b) A curve is given parametrically by. x = cos θ, y = cos(4θ), 0 < θ < π.

17 Jun 2019: y(0) = y′(0) = 0 , f (x) = 2k cos kx ,. [10]. and. y(0) = y( π. 6k. )= 0 , f (x) =. k2. 2. [10]. [You may use the identity sin(A B) = ... sin A cos B cos A sin B.].

17 Jun 2019: dx(sin x). 12 (cos x). 72. [8]. [Hint: substitute tan x = u.]. ... and v. [6]. (c) Suppose thatx = u cos v ,y = u sin v.

17 Jun 2019: a)2. |x 3| for < x < , [1]. (b)12. (sin x cos x) for π/4 6 x 6 π/4. ... B(p, q) = 2. π/2. 0. (sin θ)2p1(cos θ)2q1 dθ. [4].

17 Jun 2019: find all the solutions of this equation. [2]. 2. Differentiate. a) cos 5x [1]. ... i; ln(e); 2i; (ii)i. [8]. (b) Write down expressions for sin x and cos x in terms of eix and eix.

17 Jun 2019: 1]. (b) sin(cos x). [1]. 9. One side of a rectangle has fixed length a = 10 cm while the length of the other side, b,increases at a rate of ... 7]. (c) Find the general solution of. dy. dx+. 2. xy = x2y2 cos x.

7 Jan 2019: a) The electric field is. E = Ȧ = Aω ( sin(kz ωt), cos(kz ωt), 0). ... The magnetic field is. B = A = Ak ( cos(kz ωt), sin(kz ωt), 0).

17 Jun 2019: The inner circle, definedby τ = a, is held at Φ = 0 and the outer circle, defined by τ = b, is held atΦ = cos(2σ). ... Hint: consider α = cos θ.] [5]. (b) What restriction must be placed on p for Bp to have a cyclic subgroup C = {I,c}?Give the

7 Jan 2019: L =1. 2m(. ρ̇2 ρ2θ̇2). 12kρ2 q. (. Eρ cos θ B. ... A =B. 2. (. ρ sin θ, ρ cos θ, 0).

17 Jun 2019: Calculate T̂ and κ for the circular helix. x(t) = (a cos t , a sin t , bt) ,. where a and b are constants. (ii) Find the normal vector and the equation ... x = a cos θ , y = b sin θ , 0 θ 2π.

17 Jun 2019: y = b cos θ,. for 0 < θ < 2π. ... 0. R. 0. r2 cos θ sin θdr dθ. [4]. (b) Find the area of the finite region of the plane bounded by the line y = 2x 3and the curve y =

17 Jun 2019: θ. ]. ]. (i) Using separation of variables, find an expression for B as an expansion in Legendrepolynomials Pl(cos θ) when r > R. ... B(r, θ) =. l=0. Bl fl(s1/2 r) Pl(cos θ) ,. where Bl is a constant and fl(s1/2 r) satisfies the equation.

17 Jun 2019: i) For B = R1A with. R1 =. 1 0 00 cos θ1 sin θ10 sin θ1 cos θ1. ... ii) λ = 1 and S = 32 in the case where x y = cos 2π3.

30 Aug 2019: KN sin2 α KT cos2 α 6πµa. L)U = (KN KT )b(o ) sin α cos α. ... bL(KN KT ) sin α cos αU = (KN cos2 α KT sin2 α)b2L(o ) 8πµa3,.

17 Jun 2019: 1]. 2. (a) Calculate. sin 2x cos x dx. [1]. (b) Calculate. ... 1]. (b) Solve 1 cos 2x = cos2 x for π < x < 2π.

18 Nov 2019: 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.

6 Dec 2019: b〉 = cos(θ. 2. )eiφ/2|〉 sin. (θ. 2. )eiφ/2|〉 (10.47). as we found in the second problem sheet, where θ = cos1(a b) and φ is the azimuthalangle ... In particular, suppose a and b are unit vectors, with a b = 0 and b′ = b cos α a sin α.Then

16 Oct 2019: Ψ(r, θ) =. =0. (. A r B r. 1). P(cos θ),. ... b) Assuming the boundary condition Ψ = 1 2 cos θ 3 sin2 θ on the surfaceof a sphere of radius a, find Ψ inside and outside the sphere assuming that Ψ

13 Mar 2019: What. motivated your choice? [Suggestions: You could try ψ = cos(πx/2α) or ψ = (α2x2) for |x| < α and vanishingoutside this interval.]. ... 8. Consider the scaled Schrödinger equation. d2ψ. dx2+ λ cos 2xψ = Eψ.

17 Jun 2019: For u = ex cos y, find v and hence f(z). ... Let the energy eigenstates of even parity be given by. ψeven(x) =. . . . A cos λx B sin λx a < x < 0A cos λx B sin λx 0 < x <

17 Jun 2019: Prove the Pythagorean theorem for spherical geometry in the form. cos c = cos a cos b. ... E = (E0 cos(kz ωt), 0, 0) B = (0,B0 cos(kz ωt), 0),.