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1 Geodesics in Spacetime‣ General Relativity by David Tong
www.damtp.cam.ac.uk/user/tong/gr/grhtml/S1.html16 Oct 2021: Γ. θ. r. θ. =. Γ. r. θ. θ. =. 1. r. Γ. ϕ. ϕ. θ. =. -. sin. θ. cos. ... θ. ϕ. =. cos. θ. sin. θ. (1.11). There are some important lessons here. -
Energetics of synchronization for model flagella and cilia
www.damtp.cam.ac.uk/user/lauga/papers/189.pdf8 Jul 2021: ψ. x. (x(1)s ,y(1)s ). = b cos(x t ), (18). ... ψ. x. (x(2)s ,y(2)s ). = b cos(x t ), (20). -
2 A Quantum Particle in One Dimension‣ Quantum Mechanics by David Tong
www.damtp.cam.ac.uk/user/tong/qm/qmhtml/S2.html18 Oct 2021: The wavefunction (2.12) can be viewed as a sum of sine and cos functions and describes a complex-valued wave of wavelength. ... 2. x. The most general solution has the form. x. (. t. ). =. A. cos. (. ω. (. t. -. t. -
justaqm
www.damtp.cam.ac.uk/user/tong/aqm/justsix.pdf7 Apr 2021: x(t) = X R sin(!B(t t0)) and y(t) = Y R cos(!B(t t0)). ... AN=. g. 4r. 1 cos. sin(6.25). The resulting magnetic field is. -
February 2021 EPL, 133 (2021) 44002 www.epljournal.orgdoi:…
www.damtp.cam.ac.uk/user/lauga/papers/188.pdf8 Jul 2021: Using spherical coordinates, wefollow ref. [12], and prescribe the tangential surface veloc-ity, us, as us = (3U0/2)(sin θ β sin θ cos θ)eθ, where U0. ... For a given value ofthe dipole strength, the maximum growth rate is obtainedfor cos k = 1, i.e. -
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). -
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). -
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). -
Computing semigroups and solutions of time-fractional PDEs with error …
www.damtp.cam.ac.uk/user/mjc249/talks/GSSI_mjc_final1 Dec 2021: 29/34. Toy example. a = cosh(x), b = sin(πx) 2, ρ = tanh(x) 2, F (x,t) = cos(20t) sin(πx),. ... ρ = 1, F (x,t) = cos(πt)(24 π2(1 x2)2), y(x, 0) = (1 x2)2, yt. -
3 Structure Formation‣ Cosmology by David Tong
www.damtp.cam.ac.uk/user/tong/cosmo/cosmohtml/S3.html18 Oct 2021: cos. (. ω. t. -. 𝐤. 𝐱. ). B. (. 𝐤. ). sin. (. ω. t. -. 𝐤. 𝐱. ). (3.143). ... This form of the solution is more useful simply because it’s quicker to write exponentials rather than cos and sin. -
3 Introducing Riemannian Geometry‣ General Relativity by David Tong
www.damtp.cam.ac.uk/user/tong/gr/grhtml/S3.html16 Oct 2021: X. and. Y. at each point, using. g. (. X. ,. Y. ). =. |. X. |. |. Y. |. cos. θ. ... θ. ϕ. =. cos. θ. sin. θ. (3.111). From these, it is straightforward to compute the components of the Riemann tensor. -
Direct measurement of unsteady microscale Stokes flow using optically …
www.damtp.cam.ac.uk/user/lauga/papers/194.pdf8 Jul 2021: θ ),. ξx (t ) = ξ0 {[A(R) B(R) cos2 θ ]eiωt } and ξy (t ) = ξ0 {B(R) cos θ sin θ eiωt }. (17). B. Low-frequency limit of Lissajous figures. The ... 1 cos2θ )2 ξ2x. 2 tan θ cos φ. 1 cos2θ ξxξy 1. -
Direct measurement of unsteady microscale Stokes flow using optically …
www.damtp.cam.ac.uk/user/gold/pdfs/unsteady.pdf27 May 2021: θ ),. ξx (t ) = ξ0 {[A(R) B(R) cos2 θ ]eiωt } and ξy (t ) = ξ0 {B(R) cos θ sin θ eiωt }. (17). B. Low-frequency limit of Lissajous figures. The ... 1 cos2θ )2 ξ2x. 2 tan θ cos φ. 1 cos2θ ξxξy 1. -
Dynamics of a helical swimmer crossing viscosity gradients
www.damtp.cam.ac.uk/user/lauga/papers/201.pdf10 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. = -
Asymptotic theory of hydrodynamic interactions between slender…
www.damtp.cam.ac.uk/user/lauga/papers/196.pdf10 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. -
Cilia density and flow velocity affect alignment of motile cilia from …
www.damtp.cam.ac.uk/user/lauga/papers/184.pdf17 Jan 2021: On the other hand, along the beating direction,the minimum distance is dkðmin cos 23 cos 18Þ, where 23 degand 18 deg are the maximum tilt angles with respect to the ... erna. l flo. wu. b||. b||0. T. y=cos (θ). x. Telθ. -
J. Fluid Mech. (2021), vol. 916, A17, doi:10.1017/jfm.2021.181…
www.damtp.cam.ac.uk/user/lauga/papers/190.pdf8 Jul 2021: φ̇2{(s 1) sin φ2, (s 1) cos φ2} 1 < s < 3,(2.17). ... MPθ 1 MPθ 2. (2.35). and. MPx1 = 2Δζ81ζ‖. (4φ1 5φ2) cos θ, (2.36). -
justaqm
www.damtp.cam.ac.uk/user/tong/aqm/justone.pdf23 Apr 2021: 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. -
Part III - Classical and Quantum Solitons
www.damtp.cam.ac.uk/user/examples/3P11e.pdf20 Apr 2021: Even braver people can choose. U(φ) = 1 cos φ. This has infinitely many minima. ... U(φ) = 1 cos φ. Again, we suppress coupling constants, but it is possible to add them back.The potential looks like. -
Cop yrig ht © 201 6 U nive rsity ...
www.damtp.cam.ac.uk/user/tong/vc/benjonathan.pdf19 May 2021: ϕ=0. z=a2ρ2. z=a2ρ2. ρ dρ dϕ dz = 2π. a. b. ... Either way, the formula yields b. af(x) dx =. β. αf(x(u))dx.
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