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20 Jul 2021: M =. (f(θ). 2 i sinθ2f(θ) sinθ2f(θ)sinθ2f(θ). f(θ)2. i sinθ2f(θ). ), f(θ) =. 1 cos θ,. where θ R. (ii) Now consider an element of ... Rab = cos θδab (1 cos θ)nanb sin θabcnc,. where a,b,c = 1, 2, 3, nana = 1 and abc is the 3-dimensional

20 Jul 2021: b) Find the solution of 2u = 16r2 on the two-dimensional domain D when. ... SP dS =. VE JdV. t. V. |E|2 |B|22. dV. (). (b) Suppose J = 0 and consider an electromagnetic wave. E = E0 ŷ cos(kxωt) and B = B0 ẑ cos(kxωt) ,.

16 Dec 2021: θ. [4]. (b) For u(r,θ) of the form. u(r,θ) = rn cos(nθ) ,. where n is a positive integer, use the Cauchy-Riemann equations from part (a) and ... T [z] =12gR. θ2θ1. R2 z′2. cos θdθ ,. where g is the acceleration due to gravity.

20 Jul 2021: b). (i) Consider two states ρ1 = |0〉〈0|, ρ2 = |θ〉〈θ|, where |θ〉 = cos(θ)|0〉 sin(θ)|1〉,θ (0,π/2). ... c,d (0, 1), a < b < c < d and〈φ1|φ1〉 = 〈φ2|φ2〉 = 1.

20 Jul 2021: φ(x,y) = x cosh y sin x Ay sinh y cos x. ... f(y) =sinh[(1 i)ŷ]. sinh[(1 i)], where ŷ =. y. L0and =. (ωL202ν. )1/2. (b) Hence or otherwise, show that. u(y,t). U0=. cos ωt [cosh cos cosh

20 Jul 2021: u(t) = (sin ωt, cos ωt). (a) Derive and fully simplify the covariance kθ(t,t′) = Cov[f(t),f(t′)] between f(t) and. ... 2. )cos. (α β. 2. ). cos α cos β = 2 sin(α β.

16 Dec 2021: x(t) = (a cos t , a sin t , bt) ,. where a and b are constants. (ii) Find the normal vector and the equation of the tangent plane to the surface Sin ... x = a cos θ , y = b sin θ , 0 6 θ 6 2π.

20 Jul 2021: T(x,t) = a. [1 ebt cos. (bκx)]. 2ab. πP. 0. ev2t sin(xv/. ... 1. 2. x2. (t)2=. 1 cos(2k log t) 2k sin(2k log t) 4k28k2t.

10 Aug 2021: Consider the surface of. revolution S with points . (2 cos v) cos u. ... Your mission is to reach the point(x,y) = (B cos α,B sin α), where B > 0.

20 Jul 2021: i) Show that θ(t) satisfies the following equation of motion. θ̈ =(ω2 cos θ g. ... SP dS =. VE JdV. t. V. |E|2 |B|22. dV. (). (b) Suppose J = 0 and consider an electromagnetic wave. E = E0 ŷ cos(kxωt) and B = B0 ẑ cos(kxωt) ,.

20 Jul 2021: φ(x,y) = x cosh y sin x Ay sinh y cos x. ... f(y) =sinh[(1 i)ŷ]. sinh[(1 i)], where ŷ =. y. L0and =. (ωL202ν. )1/2. (b) Hence or otherwise, show that. u(y,t). U0=. cos ωt [cosh cos cosh