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12 Nov 2016: 8. By considering the Fourier transform of the function. f(x) =. {cos x for |x| < π/20 for |x| π/2 ,. David Skinner d.b.skinner@damtp.cam.ac.uk. ... is. fα(x) =2i. π(α2 x2)(α cosh α sin xx cos x sinh α).

20 Jan 2009: MATHEMATICAL TRIPOS PART IB. ELECTROMAGNETISM: Examples 3. 1. It is given, as a consequence of the Maxwell equation E = B/t, that. ... b. ]. sin(kz ωt), Bz =nπA. bcos. [ nπy. b. ]. cos(kz ωt),. where n is an integer and A is constant.

2 Jul 2024: b = R sin. 2. 2. = R cos. 2. If b > R, clearly there is no scattering. ... incident(r) = eikz = eikr cos. is also a solution to the Schrödinger equation.

1 Jun 2016: sin kθ0 =. n=0a(s)n Pn(cos θ0)=. n=1. b(s)n Vn(cos θ0) (3.4a). ... similar. Let. Qn(k)= π. 0sin θPn(cos θ) sin kθ dθ, (B 1).

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

17 Apr 2012: vr = rν cos φ[c(ν 1)Pν1 b(ν 1) cos θ Pν. ], ... 2.12a). vθ = rν cos φ[cP′ν1 2a csc θ Pν b.

23 Feb 2015: weak. charged current interaction). Given R(θ) =. (cos θ sin θsin θ cos θ. ), ... 4 cos θ. (2a2D v2a2D2. )},. where v = 2 sin2 θW 12,a = 12.

1 Dec 2023: pxpy. )=S(Q;B, 1). (cos θsin θ. )and. (ψxψy. )=Q. (sin θcos θ. ), ... x, y)= 13 B2a2(cos 3θ 3 cos θ, sin 3θ 3 sin θ).

1 Dec 2023: tan δ = (Ws 1) sin Φ(Ws 1) cos Φ Q, (2.18). ... Fx(δ, n, Bi) Bifx(|δ|) sgn(δ) Fz(δ, n, Bi) Bifz(|δ|) sgn(cos δ), (3.6a,b)for some functions fx and fz.

7 Oct 2008: tn. g(t) dt. If b(t) = ε cos ϑ(t) then the integral in(2.4) is of the form. ... The first step is to rewriteb(t) in the form. b(t) = 12 [cos(ω̃1t) cos(ω̃2t)],whereω̃1 = ω1 ω2 and ω̃2 = ω1 ω2.

7 Dec 2023: r3; (12b). _c % 32r3ð1þB cos 2cÞ%a. rsin c; (12c). where we recall that B = (a2 % 1)/(a2 1) A (0,1) is the shapefactor of the spheroid. ... Fig. 3A). Otherwise, it will escape due tothe fact that :r p cos c.

25 Nov 2013: v2 =g sin2 θ. cos θ. where cos θ = d/. ... the particle and deduce that the particleleaves the surface of the sphere when cos θ = 2/3.

1 Dec 2023: u= eσ cos(2πy). I0(σ)1, v=0. (3.2a,b). Given this fixed, known velocity the concentration simply evolves via the advection–diffusion equation (2.7). ... u(x, y, t)= eσ cos(2πy)Rc′(x,y,t) 1. 0eσ cos(2πs)Rc. ′(x,s,t) ds1. (4.4).

12 Oct 2020: S =. α β. β α. ββ α.  ,are given by λk = α 2 i β cos kx, and wk = (im sin kmx)Mm=1, where x = πh =. πM1. So, all ... λk(A) =λk(C). λk(B)=. 1 12 i µ cos kx. 1 12 i µ cos kx |λk(A)| = 1, k = 1.M.

25 Feb 2008: Show that. θ̇2 =6gb. a2 3b2(cos θ cos α) (1). where b is the length OX, θ is the angle between OXand the vertical and α is the maximum value ... Combining these results,. 12 (. 13 a. 2 b2)θ̇2 gb cos θ = E/M.

8 Apr 2008: In polar coordinates z = reiθ, Ar2r(B cos θC sin θ)D =0. ... Under the inversion 1/z ρeiφ = 1/reiθ this leads to Aρ(B cos φC sin φ)Dρ2 = 0which is also a circle.

18 May 2022: uU0 sin b cos h. pr0r. 2. þX1n2. sin nb cos nhpn. ... Pane, and B. Nelson, “Soft microma-chines with programmable motility and morphology,” Nat.

11 Nov 2020: x(t) = X R sin(!B(t t0)) and y(t) = Y R cos(!B(t t0)). ... AN=. g. 4r. 1 cos. sin(1.25). The resulting magnetic field is.

26 Feb 2008: constant k:. mẍ = kx. so x = A sin(. k/m t) B cos(. ... mẍ = kx cẋ F cos t. The general solution consists of two parts: (a) the particular integral; and (b) the com-.

26 Jun 2024: n(x, t) = A(k) cos (!t k x) B(k) sin (!t k x) (3.6). ... We’ll focus on the probability distribution of some scalar field (x) which, in the cos-.

26 Jun 2024: which diverges at = 0 and =. However, the numerator 1 cos has a zero when = 0 and the gauge connection is fine there. ... cos = eg/4L. So far, our discussion has been classical. Now we invoke some.

21 Oct 2022: x′ = x cos ωt y sin ωt. y′ = y cos ωtx sin ωtz′ = z (2.18). ... 12ml2θ̇2 mgl cos θ (2.39). From which we may derive Lagrange’s equations using the coordinate θ directly.

25 Jan 2007: pair abmeans add or subtract the same expression with a and b interchanged. ... cos θ =gabU. aV b. |U a||V b|. (103). If the two vectors are timelike, and both are future directed, the rapidity θbetween them is given by.

2 Mar 2004: Why?) The solution is. T =. A0 2G(1 β)1r cos θ r < a,. ... The solution is. Φ(r, θ) ={ (. 1 3 ln(r/a))r cos θ/90 r < a.

10 Sep 2021: e1 = cos χ [cos θ (cos φex sin φey ) sin θ ez ] sin χ [ sin φex cos φey], (1). ... the helix is R = sin(ψ )/(π N ) and the pitch is p = 2 cos(ψ )/N.

10 Sep 2021: helical turn and b = cos ψ. ... ez Fp = ζ T1 LT / cos ψ. 0[U (βT 1)b(R sin ψ U b)] d s. =

9 May 2023: taking the opportunity to remind ourselves of this. In Electromagnetism, the magnetic field obeys r B = 0 which means that it can be. ... magnetic field B in terms of the current density. B(x) =µ04.

7 Nov 2017: b β(b b‖) , (43). respectively, where. P (θ ) = (1 U0)b b‖ cos θ (1 U0)(b b‖) cos2 θ, (44). ... uγ̇ th =. sin2 θ 2 b‖b. (cos θ q)2, (45).

11 Mar 2010: β2π. π. 0. sin θ dθ eβµB cos θ (10.5.3). =I. ... for the 2π, and the integration over θ is doneby setting x = cos θ.

26 Jul 2013: 1(a). Our new result is illustrated in Fig. 1(c), where we showthe dynamics of swimmers undergoing reciprocal motionwith velocity Uswim UðtÞe and UðtÞ U cos!t withU 5 ... a) No swimming, (b) steady swimming at speed U 5 m=s, (c) reciprocalswimming at

3 Dec 2022: EI. ET. B. B. BR. I. T. Figure 66: Incident, reflected and transmitted waves with normal polarisation. ... E(x, t) = E(!) ek2z cos(k1z! t). B(x, t) =|k|! (ẑ E(!)) ek2z cos(k1z!

26 Jun 2024: Lint = 1. 6f2. aa@b@b a@ab@b. From this we can read o the tree-level! ... This is achieved by the so-called hedgehog ansatz,. USkyrme(x) = exp (if(r) x̂) = cos f(r) i x̂ sin f(r) (5.20).

26 Jul 2013: mal a or b at given Pe). Starting from a random initial condi-. ... For t 1, the swimming stroke,. b cos t b1 þ sin t bk; (44).

13 May 2020: 220. (|A|2 3. 4x20. ),. εv =1. 220(Br sin 0t Bi cos 0t)2. ... 1. 220(Br cos 0t Bi sin 0t). 2. =1. 220|B|2,. as required.

25 Jul 2018: 708 J. B. KIRKEGAARD AND R. E. GOLDSTEIN. Fig. 5. Drift efficiency 〈cos θ 〉 with finite tumbling time. ... 1 I1(k)I0(k). cos μ. )(〈cos θ 〉 〈q cos θ 〉) , (B.2).

26 Jul 2013: evaluated at =. In Fourier notation and given that t= cos t, they become. ... 0. = 0. 2. Pressure and stress field. We have. ̃20 =. 2De1 De21 iDe1. B. 2 cos 2 sin 22, 43. as well as zero pressure p2 = 0, so that the average

5 Dec 2008: 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 9b to evaluate the integral.

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

25 Jan 2007: with no dependence on and ) is = a br: (116)Next we have solutions, like the dipole potential, / cos , = Er cos cr2 cos : (117)What E does the rst term give? ... Hence, from (186)B = 0I4 Z 11 sdz0(s2 z02)3=2 e= 0I4s Z =2=2 cos d e; (z0 = stan)= 0I2se:

13 Mar 2008: 4 A circular helix is given by. x = (a cos t, a sin t, ct). ... Its mass density is ρ0 cos θ. By evaluating a volume integral find the mass ofthe cone.

1 Nov 2017: 3ρ̄r. (b) Consider linear perturbations about the homogeneous solution. Show that the density. ... t = B (1 cos θ). where you should determine A, B and θ.

19 Oct 2013: expected. (ii) all A’s, B’s zero except A1:. φ = A1r cos θ = A1z , u = φ = A1ez. ... z. (iii) all A’s, B’s zero except B1:. φ =B1 cos θ.

1 Dec 2023: the boundary condition to be imposed is. u=cos θ and τrθ =%Bi sgn(y) at r =1 (2.7a,b). ... balance. But FEAE′ =2FAE, where FAE =rEI Bi[2(πβ2) cos β2 sin β2] is the force on the section AE (see appendix B.1).

20 Jan 2009: Repeat the calculation in spherical geometry, i.e. for concentric spheres of radiir = a, b, b > a. ... For (iii), from a decent diagram of spherical polars, find k = er cos θ eθ sin θ. ].

1 Dec 2023: for transverse waves and. ψ =0, ψy =aπ cos πx0, on x=x0 a sin πx0 and y=0, (2.17a,b). ... 12 π. 2y= 14 π sin(2πy) cos(2πx)12 π. 2y, (3.11). Dow.

2 Jul 2024: is given by. H(t) = eE0 x cos(!t). Our goal is to find the eigenstates of the time-dependent Hamiltonian. ... The diagonal matrixelements are particularly simple. h i|H| ii = eE0 h i|x| ii cos(!t) = 0.

26 Feb 2008: Tx = T sin θ cos φ. = T (r/l)(x/r). = T x/l,. ... ζ = eiωt sin λ(A cos(. g/l t) B sin(. g/l t). ).

31 May 2024: y = r sin sin. z = r cos. In polar coordinates, the infinitesimal distance between two points can be simply de-. ... dt(r2̇) = r2 sin cos ̇2 ,. d. dt(r2 sin2 ̇) = 0 (1.12).

28 Nov 2013: X = 0 is given byF′ G = c0, (cos X σ0) G sin XF = 0, (4.4a,b). ... Θ = Ae(1κ2)τ cos κX Z, W = Ae(1κ2)τ cos κX, (5.1a,b)for any amplitude A.

11 Nov 2006: Then. z1z2 = r1r2 (cos θ1. cos θ2 sin θ1. sin θ2i (sin θ1. ... 1.30). Proof. From (1.25). cos nθ i sin nθ = ei (nθ).