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28 Dec 2021: 6. (a) Zero; (b) sin. 𝜋. 8cos. 𝜋. 4. 7. (a) sin 𝑥 sin 2𝑥 sin 3𝑥; (b) cos 𝑥 2 cos 2𝑥 3 cos 3𝑥. ... 7. (a) y= Acos 3 xBsin 3 x ; (b) y=cos 4 x.

16 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.

8 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).

18 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.

18 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.

8 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).

16 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.

5 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).

8 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).

22 Jul 2021: Threemetrics are considered. The first two are gradient based. B(c) = cos. ... Ni=1. cos. (h(c),. Fy(h; θ)h. h(i). )(7). where cos(a,b) = 1aTb/(||a||2||b||2).

15 Oct 2021: a) f(x) = sin x. b) f(x) = cos x. c) f(x) = exp x. ... f. 0(x) = cos 1. x. 2x sin. 1. x. [Hint: Use Theorems A.2, A.3].

21 Nov 2021: T2 ={(. (2 cos θ) cos ϕ, (2 cos θ) sin ϕ, sin θ). : ... Nowconsider the subset A = {(0, 0, 1), (0, 0,1)} of the two-dimensional sphere S2, and the subsetB = {(2 cos θ, 0, sin θ) : θ [0, 2π]} of T2.

21 Nov 2021: T2 ={(. (2 cos θ) cos ϕ, (2 cos θ) sin ϕ, sin θ). : ... Nowconsider the subset A = {(0, 0, 1), (0, 0,1)} of the two-dimensional sphere S2, and the subsetB = {(2 cos θ, 0, sin θ) : θ [0, 2π]} of T2.

7 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.

8 Feb 2021: φ(s,v) = α(s) r(n(s) cos v b(s) sin v). where n = n(s) and b = b(s) are the normal and binormal vectors of α. ... φ(u,v) = ((2 v sin(u/2)) sin u, (2 v sin(u/2)) cos u,v cos(u/2)).

8 Aug 2021: Replace‘Similarly, we can can take a = sin φ, b = cos φ for some real φ.’ by. ... Similarly, we can can take c = sin φ, d = cos φ for some real φ.’.

15 Oct 2021: that we have. θ. ϕ1(θ,φ). ' eθ :=.  cos θ cos φcos θ sin φsin θ. ...  , φ. ϕ1(θ,φ). ' eφ :=.  sin θ sin φsin θ cos φ0.

15 Oct 2021: Res[F,i] = cosh(π θ)4 sinh π. , Res[F,n] =cos(nθ). 2π(1 n2). ... b) Suppose φ E (Rn), show that. hi φ Diφ, as h 0,.

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) ,.

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.