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Unformatted text preview: MAT 362 Answers to selected exercises, week of Oct. 12 Graded problems 1. We have F = r / bardbl r bardbl 3 , which is undefined at the origin. The divergence theorem cannot be applied because F is not C 1 on the unit sphere. The flux integral must be evaluated directly. On the unit sphere, the normal vector is r (i.e., any radius), which may be verified readily by parametrizing the sphere in polar coordinates and computing T θ × T φ directly. Thus, iintegdisplay S F · d S = iintegdisplay S r bardbl r bardbl 3 · r dS = iintegdisplay S bardbl r bardbl 2 bardbl r bardbl 3 dS since r · r = bardbl r bardbl 2 = iintegdisplay S 1 bardbl r bardbl dS = iintegdisplay S 1 dS since bardbl r bardbl = 1 on the unit sphere = surface area of the unit sphere = 4 π. 4.1.8. The goal here is to find the right coe ffi cient that goes in front of the integral in the analogues of Eqs. (4) and (6). The idea is to get normality: we want the coe ffi cient c that makes c integraldisplay T / 2 T / 2 cos 2 parenleftBigg 2 π kx T parenrightBigg dx = 1 . (That is, we want ( cos(2 π kx / T ) , cos(2 π kx / T ) ) = 1.) We have integraldisplay T / 2 T / 2 cos 2 parenleftBigg...
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 Fall '06
 CALLAHAN
 Trigonometry, Advanced Math, Fourier Series, Cos, Periodic function, dx

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