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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

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),.

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

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,.

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: 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: 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.

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: 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),.

30 Aug 2019: η2 = 2cos θ1 sin θ1. cos θ2 sin(θ1 θ2)h0,. where θ1 and θ2 are the angles between lines of constant phase and the vertical inthe two regions. ... You may assume thath = h0 at x = 0 and that sin θ >. 12E cos θ. Find expressions for h(x) and

30 Aug 2019: 3)(f) Assume f(t) = A1(t) cos(ω1t) and ω = ω2 ω1. ... Show that exp(iθA) = cos(θ)I i sin(θ)A for θ R. (b) Use the result from part (a) to show that.

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.

17 Jun 2019: Hence express y2(x) in terms of y1(x) and W(x). (b) Verify that y1(x) = cos(xγ) is a solution of. ... y. 0Q(x,y′)dy′. or otherwise, show that V is conservative. (b) Now let V =(. 1 cos(2πx 2πy),2 cos(2πx 2πy)). Show that there existsa smooth

17 Jun 2019: 1. c22E. t2 2E = 0 and E = 0 where c =. 1. ǫ0µ0. (b) Suppose that E = E02(1,1,0) cos(kz ωt) where E0, k and ω are real constants. ... f(u). (b) Now take f(s) = cos s 2, g(s) = sins, 0 6 s < 2π.

17 Jun 2019: a) Show that (r/x)y = cos φ and find a similar formula for (r/y)x. ... K(r, θ) =1. 2+. n=1. rn cos nθ,. and for an arbitrary 2π-periodic function f, let.

6 Nov 2019: Define a new kernel on the interval [a,b] by,. k̃(M)(x,x0) = (b a)M. ... In particular they defined inducingvariables such that. cov(um, f (x)) = cos(wmx), (2.27).

17 Jun 2019: π. n=1. cos[(2n 1)x] cos[(2n 1)ct]. (2n 1)2. (b) Determine a periodic function P(x) such that this solution may be expressed as. ... d) Suppose that E = E0(0,1,0) and B =E0c (0,cos θ,sinθ) where 0 6 θ 6 π/2,.

19 Jul 2019: 161 –. by a sign, since this can be compensated by the cos function. ... Insisting that the two. cos functions agree, up to sign, gives us the condition.

17 Jun 2019: S3 =. 2. (1 00 1. )and. B(t) = B(sin α cos ωt, sin α sin ωt, cos α). ... eiωt/2 χ+ i. (ωλ. sin α sin(. 12λt. ))eiωt/2 χ. where λ (ω2 2ωB cos α B2)1/2 and.

17 Jun 2019: has polar equationAr = 1 ε cos θ with A > 0 and 0 < ε < 1.]. ... i) For B = R1A with. R1 =. 1 0 00 cos θ1 sin θ10 sin θ1 cos θ1.