Question

good evening.

i need help with this calculus question.

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or more conveniently, Given the vector field in space F(x, y, z) = ri+yj + zk (x2 + y2 + 22)3/2 F(r) where r = ri+yj + zk and r= ||1|| = r3 r 22 + y2 + 22 (instead of p) (a) [10 pts] Find the divergence of F, that is, V.F. (b) [10 pts] Directly evaluate the surface integral F. NdS where S is the unit sphere z? + y2 + x2 = 1 oriented with outward normal vectors. (c) [5 pts] Do these two results from (a) and (b) contradict the Divergence Theorem? Explain.

Answer 1

xi + y ) + 2k = ( ر) F .راح پ" + 1) where r= xi + ijt zů. : = (۲) م . لارو-x = || || -۷ ( از اداره امور ) و ( پردي و عوده ) ) ۲۶۰ ع | + ** » Now, en ( روہ 22 رهيو ) و - .يو) ,3 ( او و ۹۴) .رويد 7 + 8 x 3 .3 ر و +2) ** + +2 ) . 31 - 1912 و ** -2 . ( 174 ) % ۸۶+y42) %. sly 1 - 21 او 2 - ا ي * % (-ړې وه ) * ؟ (2+7+8) . (خودا و چ دو۔ والا پاره 24) . (22 -و +1) + (27-29 794 ) + (27 - +9) ( 42 روبه) . - : = F2 . So, divergence of f ( 2 + (وہ ) 2 (1-" ہوا ) v (ما) || (د " )Y 2 x+yutar (87+7+ 2 -1) ام زود به 492 .(ا-ه 12هه ||

andy --2 Z +72 F. Nas=2. F.N dady. 21. Tả.kl artytz 255. atys (nty +24% (uty +2 ) Ž andy. SS. z (n4yº+zy). 3+ y2, =2SS andy. artyl - 2 2. rardo. 22. s nar Crie Yersino . S. -r VD 020 an ir dr. 2.2. s s 623 20 rar. 47. 723 2-2 ñ S' s' d (1-r) [-***** - 90 [11-voy3] = -4 0-1 Y20. 2.--2 ñ 2 41 - SSF.nasa (C) Now, S55 7. F du so. By Divergence theirew, SSF Nids = BIS Vifd V. Here. SS F-Nas=arto Sllaron. So, it contradicts the divergence therren?
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