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Mathematical Methods I Natural Sciences Tripos, Part IBDr Gordon ...
www.damtp.cam.ac.uk/user/gio10/N21b.pdf5 Dec 2008: 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 ... 1 e cos φ,. where (ρ, φ) are plane polar coordinates and e is a -
MATHEMATICAL TRIPOS PART IB ELECTROMAGNETISM Examples 2 1. A ...
www.damtp.cam.ac.uk/user/ngb23/EM/B10b.pdf20 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. ]. -
Part III Applications of Differential Geometry to Physics, Sheet ...
www.damtp.cam.ac.uk/user/md327/PartIII_geom1.pdf22 Jan 2020: Consider the matrix representation of the Euclidean group E(2) in twodimensions cos θ sin θ asin θ cos θ b. ... a cos ψ sin θ. b+ sin ψ. 1. l. θ. -
A5a.dvi
www.damtp.cam.ac.uk/user/hinch/teaching/A5a.pdf13 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. -
J. Fluid Mech. (2021), vol. 914, A20, doi:10.1017/jfm.2020.1112 The…
www.damtp.cam.ac.uk/user/lauga/papers/186.pdf8 Jul 2021: Γ ={(r,θ): r = R(θ) = b cos θ. c2 b2 sin2 θ. = ... 2. ξ2+ η2. )and h = a. cosh η cos ξ ,(4.33a,b). -
MATHEMATICAL TRIPOS PART IB ELECTROMAGNETISM: Examples 3 1. It ...
www.damtp.cam.ac.uk/user/ngb23/EM/B10c.pdf20 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. -
Michaelmas 2016 1B Methods 1B Methods – Example Sheet ...
www.damtp.cam.ac.uk/user/dbs26/1BMethods/probs3.pdf12 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). -
A3a.dvi
www.damtp.cam.ac.uk/user/ngb23/pdfs/A3a.pdf20 Apr 2012: ex2. sin 2x). ,. (ii)d12. dx12(x cos x) using (a) the Leibniz rule and (b) repeated application of the product rule,. ... i) (x2 a)3/2,. (ii) ln(cos x),. iii) exp. {. 1(x a)2. }. ,. -
J. Fluid Mech. (2021), vol. 914, A20, doi:10.1017/jfm.2020.1112 The…
www.damtp.cam.ac.uk/user/gold/pdfs/worms.pdf5 Mar 2021: Γ ={(r,θ): r = R(θ) = b cos θ. c2 b2 sin2 θ. = ... 2. ξ2+ η2. )and h = a. cosh η cos ξ ,(4.33a,b). -
PHYSICAL REVIEW E 93, 043125 (2016) Rotation of slender ...
www.damtp.cam.ac.uk/user/lauga/papers/119.pdf1 Oct 2016: L1. L1{L2, s1 sin ωt,s1 cos ωt} ds1. L2. L2{s2,0,0} ds2. ... r1 r = {αs,b sin(kx ωt ),b cos(kx ωt )}, (21)r2 r = {b cos(kx ωt ),b sin(kx ωt ), αs}, (22). -
A5b.dvi
www.damtp.cam.ac.uk/user/hinch/teaching/A5b.pdf13 Mar 2008: F = (3x2yz2, 2x3yz, x3z2), G = (3x2y2z, 2x3yz, x3y2),. H =(. 3x2 tan z y2exy2. sin y, (cos y 2xy sin y)exy2. , ... u = ((x cos y cos y y sin y)ex, (x sin y sin y y cos y)ex). -
A reciprocal theorem for boundary-driven channel flows
www.damtp.cam.ac.uk/user/lauga/papers/111.pdf31 Dec 2015: ex. ez. ) a2 y(ez. ey eyez). (16). The contour of the elliptical cross section can be parameterized as x = a cos , y = b sin , with0 2. ... a sin e. x. b cos eypa2sin2 b2cos2. , n =b cos e. -
J. Fluid Mech. (2018), vol. 856, pp. 921–957. c© ...
www.damtp.cam.ac.uk/user/mt599/papers/2018-jfm.pdf10 May 2022: µs(θ)=. l=0. µslPl(cos θ), µp(θ)=. l=0. µplPl(cos θ). (2.8a,b). In the case of our spherical particle, the slip velocity due to either one ... s0 =sb(rc)r sR3. 2r2(sz cos θ s. y sin θ sin φ). -
Hydrodynamic synchronization in strong confinement
www.damtp.cam.ac.uk/user/lauga/papers/187.pdf8 Jul 2021: φ̇n = ω0 βω0 (cos χn cos χn1 ) βτ ω20 (sin χn1 sin χn ), (16). ... θ1 sin θ2 ) sin α1 sin δ cos θ1 (sin θ1 sin θ2 )]}, (46). -
Coherent Field and Specular Reflection At Grazing Incidence on a…
www.damtp.cam.ac.uk/user/ms100/PAPERS/JASA94spec.pdf13 Dec 2018: b/ø (x,h) = exp{ik [xS h (x) cos 0 ] }. •ei•$[ 1 ikh(x)cos O- ( k2/2 )h2(x )cos 2 0]. (9). ... k2 •b• ø (x,zt) = - 1 ikh (x) cos 0- • cos 2 0 X [h2(x) (Zl-h) 2] )e ikxs -(za-h) •xx ikcos0. -
J. Fluid Mech. (2018), vol. 856, pp. 921–957. c© ...
www.damtp.cam.ac.uk/user/lauga/papers/152.pdf19 Oct 2018: µs(θ)=. l=0. µslPl(cos θ), µp(θ)=. l=0. µplPl(cos θ). (2.8a,b). In the case of our spherical particle, the slip velocity due to either one ... s0 =sb(rc)r sR3. 2r2(sz cos θ s. y sin θ sin φ). -
Chapter 11 Phase-Plane Techniques 11.1 Plane Autonomous Systems A ...
www.damtp.cam.ac.uk/user/reh10/lectures/ia-dyn-chapter11.pdf26 Feb 2008: e1 B. (cos(βt) i sin(βt). )e2}. = eαt(a cos(βt) b sin(βt). ... J =. (0 1. (g/l) cos θ 0. ). The phase diagram has this form:. -
Lect16.dvi
www.damtp.cam.ac.uk/user/na/PartII/Lect16.pdf21 Feb 2005: sin(θ ψ) sin(θ ψ) = 2 sin θ cos ψ. We have. ... ρ(B) = cos. (. π. m 1. ). 1 π2. 2m2,. -
mf4.dvi
www.damtp.cam.ac.uk/user/tong/dynamics/mf4.pdf16 Nov 2005: c) P = 2. q(. 1. q cos p). sin p and Q = log(. ... q1 = Q1 cos λ P2 sin λ , q2 = Q2 cos λ P1 sin λ. -
http://journals.cambridge.org Downloaded: 24 Aug 2009 IP address:…
www.damtp.cam.ac.uk/user/hinch/publications/JFM56_803.pdf24 Aug 2009: he, = sin 0 cos 8 ( A 2 - sin2 0)B sin 0 cos $, he, = - sin2 8 ( A 2 - sin2 0 ) t cos 0 cos $, he, = (A2 - sin2 8)j ... Gal. ( 6 b ). ( 6 c ). ( 7 ). and the orbit equation (1) reduces to. 4 = [ C O S 0 (A2 - sin2 6) cos 91 with 0 constant.
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