2)
For the triangle, we have
3)
From the definition of the sine, we have
And for the right angle triangle, we have
4)
From the definition of the tangent, we have
And for the right angle triangle, we have
And so,
And from Pythagorus theorem, we have
Q. 1 (a) Apply the lowering operator L- to Y22 (θ,φ) to find Y21 (θ,φ). (b) Apply the raising operator L+ to Y30 (θ,φ) to find Y31 (θ,φ).
2. (a) Given that (3 cos2- 1) find Y2-1(θ, φ) by direct differentiation using the lowering operator . (Ans: 15 (b) Show that Y2,-1(0,) is normalized, that is (c) Show that Y2,0 (θ,d) and Y-1(0.0) are orthogonal to each other. 2. (a) Given that (3 cos2- 1) find Y2-1(θ, φ) by direct differentiation using the lowering operator . (Ans: 15 (b) Show that Y2,-1(0,) is normalized, that is (c) Show that Y2,0 (θ,d) and Y-1(0.0) are orthogonal to each other.
х у V 1.0 7.6 1.0 1.0 2.0 8.7 11 2.0 2.0 3.0 7.3 3.0 3.0 4.0 5.8 11 10- 9- 8 7-1 6 5- 4+ 3+ 2 4.0 4.0 11 10 9+ 8+ 71 6+ 5+ 4+ 3+ 2+ 1. X 5.0 8.2 5.0 5.0 6.0 4.9 6.0 6.0 X 7.0 4.5 7.0 7.0 8.0 7.2. 8.0 8.0 0 1 2 3 4 5 6 7 8 9 10 11 7 8 9 10 11 9.0 5.9 9.0 9.0...
Find the solution to ∇2 Ψ(r, θ, φ) = 0 inside a sphere with the following boundary conditions: ∂Ψ (1,θ,φ)=sin2θcosφ.∂r Find the solution to V2(r, e, p) =0 inside a sphere with the following boundary conditions: ay (1, e, p) sin20 cosp. ar Find the solution to V2(r, e, p) =0 inside a sphere with the following boundary conditions: ay (1, e, p) sin20 cosp. ar
4. (a) Three point masses are attached to a massless rigid rod. Mass m,-1.0 kg is located at x = 1.0 cm, mass m2-2.0 kg at x = 2.0 cm and mass m,-3.0 kg at x-3.0 m. Find the center of mass of the system. (b) Find the center of mass of the four masses as below. mi 2.0 kg at point (1,2) cm; m 3.0 kg at point (2,-3) cm; m -4.0 kg at point (3,-4) cm and m...
16) For the circuit shown in the figure, determine the current in (a) the 1.0-2 resistor. (b) the 3.0-Ω resistor. (c) the 4.0-Ω resistor. 2.0 Ω W 3.0 Ω 4.0 Ω, 12 V 5.0 Ω ΜΜΜ, 1.0 Ω
Consider the following surface parametrization. x-5 cos(8) sin(φ), y-3 sin(θ) sin(p), z-cos(p) Find an expression for a unit vector, n, normal to the surface at the image of a point (u, v) for θ in [0, 2T] and φ in [0, π] -3 cos(θ) sin(φ), 5 sin(θ) sin(φ),-15 cos(q) 16 sin2(0) sin2(p)216 cos2(p)9 3 cos(9) sin(9),-5 sin(θ) sin(9), 15 cos(q) 16 sin2(0) sin2(p)216 cos2(p)9 v 16 sin2(0) sin2@c 216 cos2@t9(3 cos(θ) sin(φ), 5 sin(θ) sin(φ) , 15 cos(q) 216 cos(φ)...
3.58 (a) If U(x, y, z) = xy72, find ▽U and V2U. (b) If V(p, φ, z)- P (c) If W"(r, θ, φ.)-z? sin θ cos φ, find W and VzW. sın, find wandV2V.
A variety of spectra for an organic compound with molecular formula C10H16O are presented below. The experimental accurate mass using (+) APCI source is 153.1280 u. The 1H, 13C, COSY, HSQC and HMBC NMR spectra are given in the following slides. Propose a structure for this unknown and answer or address the following questions or requirements: a. Using the most abundant isotopes of C, H and O, what are the errors in ppm and milli-Daltons for the experimental accurate mass?...
4.0 m-3.0 m |j + 1.0 m k and b-Ί-1.0 m-1.0 m-4.0 m |k . In unit-vector notation, find (a) a + b Two vectors are given by a = (b) a-b,and (c) a third vectorç such that a-b+(10 (a) Number k Units (b) Number k Units (c) Number 3 Units