Consider the sheet formed by the intersection of the curves: x = 0, x = 4, y = 0, y = 3 [=] cm, with a variable density of mass per unit area ρ(x,y) = xy [=] g/cm2 . Write and evaluate multiple integrals to calculate the following: a. The area of ​​the sheet [=] cm2 . b. The mass of the sheet [=] g. c. The shell moments about the x & y axes (Mx & My) [=] g∙cm.

B2

Q3. For a system at T = 300 K, what has to be the value of (ɛ - u) so that the distributions of Fermi- Dirac, Bose-Einstein, and Boltzmann agree to within 1%? Is this condition satisfied for gases in the atmosphere?

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Answer in 90 minutes please. Count as double the questions if necessary.

2. Spinors and rotations (a) For a function f(ø) that can be expanded in a Taylor series, show that f(o +4) = e²L±p/hf(6) where p is an arbitrary angle. For this reason, Lz/ħ is called the generator of rotations about the z-axis. More generally, L.ân/ħ is the generator of rotations about the direction în, in the sense that exp(iL.ng/h) effects a rotation through angle y (in the right hand sense) about the axis în. In the case of spin, the generator of rotations is S.în/ħ. In particular, for spin 1/2., X = ei(oî)g/2x tells us how spinors rotate. (b) Construct the (2x2) matrix representing rotation by 180° about the x-axis and show it converts spin up (x+) spin down (x-) as you would expect. (c) Construct the matrix representing the rotation by 90° about the y-axis, and check what it does to X+· (d) Construct the matrix representing rotation by 360° about the z-axis. Discuss the meaning of the answer. (e) Show that ei(o-î)e/2 = cos(p/2) + i(în · o) sin(4/2)