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M402-ex2.F07.solns

# M402-ex2.F07.solns - or 1 1 Fill in the blanks with the...

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Unformatted text preview: or] 1 1. Fill in the blanks with the best, simplest answer. Carry out all operations. it is not necessary to show work; however, some partial credit will be given if the work is shown on this page. in the following r is the position vector, xi refers to x, y, or z, and q' refers to the three variables in the general coordinate system. The r, 8 and {p are coordinates in the spherical system; f‘, 6 and o are the corresponding unit vectors. F and f are function of x, y, and 2. S is a vector space; p, w, and t are (non zero) vectors in the space S. (1 and [3 are (constant) complex numbers and a and b are (constant) real numbers. Bold face type indicates a vector and operators act on all functions to the right. a, b, and k are constant vectors. dV E d3x, dV'E d3x' 4 points each, except as marked. Note that i = \/-l - u - u : _K — Vi a) Estimate 0n[ tr - k 10 '47 |/1r| ] using a Taylor series: — to“ EFL: RM {"1" :Q‘J‘y M, if“ LAIer ,: anal «‘t r: .. Iv ' v t‘ r”- b) The coordinate system s,t,w (5 q‘,q2,q3) has a metric tensor given by gm.l : 8mm nq" mq'11 (no sum on n or m): In the S,t,W system the differential volume element is : b 5+!» (5 SdtcluJ a} : Z 3112.?) I ﬁt: 55”” o o M; - Hf 2 2 {5/ c) For the stw system 1n b), a“ Smpm s t (w+rt)5(-nw)5(s -2)5(1-5t ) dsdtdw = as t m [25! let A "- -— __ r L _ _ ‘ " .E a d) (L X L) y = J“?- L X”? “ 3 3‘] (hint: See Lon front page) LXLj Ln}: ' I? (it? V 2 71‘“ e“at “nova :73 e‘taQix; .— tﬂ’ix -3212] 3 WW" “4‘5 ixJVK e) ffover the Open surface r : 712 20 [V X ln(r)¢ ] - dA = “4 TFvQ/vﬂ ((0 is f‘ x 6 in spherical coordinate system) gyma) @wvctmdtad) f) The directional derivative of In [(r x k) 'a] along 2 = m) r i * .ipld%—?§- ‘i‘ﬁm v. fffallspace[a/r'r|2]_dv=_ o A k Tu? 503% Fé’l‘ "S— L h) fr:22 tor r:4y+32 dr'virexp('irl)]: 13: :R‘iqdﬂge " age ] szfSletF" ‘ . fffallspaceCOS( k ' I“) v I - r1l-1 (:13)('I Z + 4-". R ' j) [cxp(0.05’l- V)] r2 = W + 0-0521 % (an? :36?» La 2) J _ k) (270-2 fffall 1" space fffall kspace (rt r) expiik '(ri' 32 )]d3k dBX’: 2“ Chg-“P : frail: 1) V 2 [ sin( 5r-k)]= — case; so: 533-?) m) Iii-over sphere ofradiusz V2 ir ' 7Iii-l d3X : O n) 1f dr/doF n y and G(r) : sin[k- r] then d/dco on) = T?“ V ‘P (Cc-.0 \$2.?) A V‘ o)r-(r-V)V[HIrt]= 1 ’F——.— tiili: =Y‘ iii") 2 2. Set up, but do not integrate the solid angle subtended by that portion of the cone surface y2 = X2+Z2 which is enclosed by the surfaces (x-l)2 + z2 = 16 and y = 0. Assume that the viewing point is at r = 69-2 . Determine all limits and eliminate all but the variables of integration. Put integrals in terms of integrations in the x,y,z coordinate system given. Sketch the projection area and show all your work. Hint: project the surface onto the xz plane. Y1: X14 f“?- r .1. ayzznz => 3% ‘ 1-1 53; a A A v t1 : ~1‘I‘I +2)“ + .m “MEN : ONT-r'il-‘W SE £2.35AA .31 = _ .r 3 Invala Q: (EM 3x4} [x‘+L~/»L\‘+L2+I)‘l3" WM! 3x2 l - 1 l H w t + [Am vzvb/ow‘rziﬁml ' Y L 7' _“,-____.___/ v 7 = H? h : A? 1 1. 3A 2 S [Xi-F (V13 + (“3] ‘ Y 3. Assume that F(r) satisﬁes the following conditions for all r: V ° = p0 8(1’2-32) 5(COS 8(2(p-TE) where p0 is a constant. V x F(r) = 0 Use the Helmholtz theorem to ﬁnd an F(r) which is ﬁnite for all 1'. Be sure to state the general form for F, and do all the integrations. Show all work. Ear) = JV]; +5xK , AR] :0 4%; = J\ CL) 43,: W lam \ (3° SEQ-02) Scmé‘)5(1q’l‘") rulr‘ane‘ée'd‘e' 1 '5 Lnr 1 NH MN, MW“ Era wtaumL u u . L.- uz (_Ju>6)4(f' - ,L Po , 5(v-a)+S( m scope) am In)» AN ‘ ‘m nm' {ind Mi & p‘2a. (“a L L'A“ "L’T e‘:’ll'2. ‘H‘I’ Li \F-owl Q.>TVZ _ | (inc-37]) .. a _ of , __\____,_ = ‘6” - A 1 PM - V P 7% (\Ra‘li mm ...
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• Fall '09
• J.
• Geographic coordinate system, Coordinate system, Spherical coordinate system, Polar coordinate system, corresponding unit vectors, space fffall kspace

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M402-ex2.F07.solns - or 1 1 Fill in the blanks with the...

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