Question

(20pe - 2) – 20e -22 = 0 showing that G has no outward flux. Hence, (VG) dy = 0 PRACTICE EXERCISE 3.7 Determine the flux of D = p cos a, + z sin an over the closed surface of the cyl- inder 0 z1, p = 4. Verify the divergence theorem for this case. Answer: 647.

Answer 1

/ Determine the flux of D = p² cos 4 a, 4 Z Sing an over the closed surface of the Cylinder oszc1 P=4. Verify the divergence theorem for this case Give vector D is in Cylindrical coordinate system. Flux= Amy :-) ... I F. ñ ds And divergence theorem: os Fañ ds = for du Z S2 Given surface is Cylinder of oczet and f= 4 We divide whole surface into three parts. S: (Z=0), S, (2-4) and S3 (8:4) S3 Si

F. ñ ds = JF. ñds + Sends + Seads + Spuñas 'S SA S in our vector is D=p² cos²&ap + 2 Simø ad to a on S1 : 2=0, ds=dsdedd, m =- az hence D. ñ=0 - =) Doñds =0 S ño E a₂ on Sq: 2=1, ds: gde dø hence Doñ=0 Sonds=o =) )مہ on Sz: 8=4, ds=4 dødz

Ô is unit vector mormal to surface. let fif-420 ñ = geadf grad Gradient in Cylindrcal coordinat If= df df df ар + ӘР s do + 1 az dz hence to a If = 1 as to a 1 10f1= Juaget (o)+(0) = 1 DA a, 101 p² cos²0 = D. ñ = our s=4 = D. 16 Cos?! Sonds I recente 16 cos²u. a do dz S;

2 T =64 s cos? $ doo + 2 T S 1 + cos20 de = 64 2 2T 64 Sim20 U + 2 saja: 40-0-] 64 T Dia ds = S₃ Join hence & Dinds=fonds + finds+fonds ose SA otot 64 T - So giôds = 647 which is Rocuited flux.

To verify divergence theorem Now we calculate IV.D du where du= 8 d z df dø dup 1 dua + duz + 1 + up = divergence in Cylindrical Coordinate system vūs Ja РОФ f I. D = 2 (p² cos²0) + 1 d (z Simp) + 2 (0+1 (płosy dp Juz SZ f. do Р 2P cos? ¢ ¢ 1 2 caso tecos²p I 3p cos²a + 1 z cosa - s 2T 4 1 Lov voodu= f cos26 + 1 zloso f a 0 0 ss Soles cosesini • Jedz dras i s [ 3pce [ 3p cor $27 bis zesø Tårds 2T 4 ono 3 pros2 & + 2 cosa 7 de do 2p o

- coso ] do do o 2T 4 p² cos²&t f cose do 2 O 21 64 cos²o + 2 cose SI + 2 cosø J dø - [ copos tatua) a acusa Jolie [ S* ( * ) 2544 **(**) *- 20 271 0 - too 64 T Juodu honce & Dim ds = / J. o du Diverg rgence theorem is verified.