BIRLA INSTITUTE OF TECHNOLOGY, MESRA, RANCHI DEPARTMENT OF MATHEMATICS MA Egieeig Matheatis MO/7 Tutoia Sheet No. Modue IV:. Defie Beta futio ad Gaa futio.. Pove that,,,. Pove that, d. Pove that. & whee og q q d q 5. Pove that, ; og d 6. Pove that, 7. Show that whee is ositive. 8. Pove that: os si / d ad hee fid the vaue of. 9. Assuig, ; si d ove that ; si. Show that : i d ot d ta ii d = 6 5 iii / 9 6 8 d iv a d e a
v e d = vi d vii Mutie Itegas: / 6 si d viii / si 5 os d 8 5. Evauate: a d d b e / dd. Evauate: a dzdd / sev t b e ddtdv. Chage the ode of itegatio of the foowig doube itega: a a dd a a a b a f, dd. Chage the ode of itegatio of the foowig doube itegas ad hee evauate the sae: a e dd a b a a dd dd d e / dd e e dd f si dd 5. Tasfo fo Catesia to Poa fo ad the evauate: a dd b dd 6. Evauate dd ove the oo of the eisate a os. a 7. Fid the aea that ies outside the ie ad iside the Cadioid os. 8. Fid the aea of the egio oo to the iteios of the Cadioids a os ad a os
9. Fid the voue oo to the two ides ad z.. Fid the voue bouded b the ide z 6 ad the aes ad.. Fid the voue of the otio of the shee z a ig iside the ide a.. Fid the voue bouded b the aabooid az, the ide a ad the ae z =.. Pove that the voue eosed b the ides a ad z a. /. Evauate e dd usig the tasfoatio u = + ad v = -. 5. Evauate + dd, whee R is the aaeoga i the -ae with veties,,.,,,. usig the tasfoatio u = + ad v = -. 6. Fid b tie itegatio the voue of the eisoid z. a b Modue V: 7. Show that the oditio that the staight ie a os bsi a touh the ie k os is b k ak. 8. Wite dow the oa equatio of the ie of adius uits with ete o the iitia ie at a distae of uits fo the oe o the ositive side. 9. Fid the oa equatio of eah ois whee the eetiities of ois setios with oe fous at the oigi aog with the dieti oesodig to that fous ae give: i e /, os ii e, si iii e /, os iv e /, os. Fid the eetiit, the dieti, vete ad ete fo the eise i aoiate oa oodiates of ois setios osideig oe fous at the oigi: i 6 ii os os iii 5 iv 5os si
. Show that the oa equatio of the eise, if the oe be at its ete ad the 9 ositive dietio of the ais be the dietio of the oa ais is 6 9 5os. Show that the equatio of hod of the oi eos joiig the oits A ad B whose oa ages ae A ad B is eos se os. Show that the equatio of taget to the oi eos at A is eos os. Show that the staight ie Aos Bsi touhes the oi eos if A e B. 6 5. Veif whethe the ie os 5, is a taget to the oi os, whee ta. If so, fid that oit. 6. Show that the oditio that the staight ie a os bsi a touh the oi eos is a e b. 7. Show that the equatio of hod of otat of the tagets to the oi eos, daw fo the oit, is eos eos os 8. Show that the dieto ie of the oi eos is e e os. 9. Show that the ous of the oit of itesetio of a ai of eediua tagets to the oi eos eesets a ie dieto ie.. Pove that the equatio of ai of astotes to the oi eos is
e e os Modue VI:. If A 5t i t j tk d i A. B dt e si. ad B si ti os tj 5tk, fid d ii AX B. dt. Coside the uve give b the aaeti equatios: ost, sit, z t / fo t. Fid the taget veto ad egth of the taget veto.. Coute the uvatue of the uve havig aaeti eesetatio: os t t si t, si t t os t, z t fo t.. Coute the vetos Tˆ, Nˆ & Bˆ fo the foowig sae uves. Aso fid the uvatue ad tosio : i ii iii iv t t R t iˆ ˆ j R t si tˆ i os t ˆj tkˆ t t R t e os tˆ i e si t ˆj kˆ R t osh tˆ i sih t ˆj tkˆ 5. Coute the tagetia ad oa ooets of aeeatio fo the sae uves without gettig T & N at the give vaue of t: i ˆ R t t ˆ i t j t kˆ, at t ii ˆ ˆ R t i t j t kˆ, t iii R t os t t si tˆ i si t t os t ˆj tkˆ fot. t t iv ˆ t R t e os tˆ i e si t j e kˆ, at t 6. Fid the adia ad tasvese ooets of veoit ad aeeatio fo atie ovig aog a uve i the oa oodiate ae. 7. Fid the adia ad tasvese ooets of veoit ad aeeatio fo atie ovig aog a sae uve i the idia oodiates. 8. Fid a veto oa to the sufae at the give oit: 5
i f, ii f,, z z z ta z, at,, iii f,, z e os z si, at,, / 6 9. Fid the ostats a & b so that the sufae a bz = a+ is othogoa to the sufae -z+z = at the oit,, -. 5. Fid the dietioa deivative of the futio at the give oit P i the dietio of the veto A : i f,, P 5,5, A ˆ i ˆ j ii f,, z e os z, P,,, A ˆ i ˆj kˆ 5. Fid the dietio i whih the futios iease ad deease ost aid at the give oit P. Fid aso the dietioa deivative of the futio i that dietio: i f,, P, ii f,, z 6z P,, 5. I what dietio fo,, - is the dietioa deivative of = - aiu? Fid aso the agitude of this aiu. 5. Evauate div R ad Cu R ad div u R whee 5 a R z ˆ i z ˆ 6 j z kˆ b R iˆ z e ˆ j e kˆ Modue VII: 5. Fid the wok doe i ovig a atie oe aoud a ie C i the XY ae, of the ie has ete at the oigi ad adius ad if the foe fied is give b F zˆ i z ˆj 5z kˆ. 55. If F iˆ ˆj evauate F d. whee C is the uve i the ae,. C 56. Fid the iuatio of F aoud the uve C whee F = i + zj+k ad ad C is the ie + =, z = 6
57. Fid the wok doe b the foe If F zˆ i z ˆj kˆ b ˆ R t i t ˆj t kˆ fo t tot. atig aog the uve give 58. Show that F = -z i+ j+ k is a osevative foe fied. Fid the saa otetia. Fid aso the wok doe i ovig a objet i this fied fo, -, to,,. 59. Aig Gee s theoe to evauate{ e si d e si d} whee C is the eise 9 6. 5 6. Aig Gee s theoe to evauate{ d e d} C C C whee C is the ie 5. 6. Veif Gee s theoe { d d}, whee C is bouded b the uve = ad =. 6. Fid the sufae aea of the otio of the ide z ig iside the ide. 6. Fid the sufae aea of the otio of the shee z 9 ig iside the ide. 6. Evauate zj S z zi k ds whee S is the sufae of the shee a i the fist otat. 65. B Gauss s Divegee theoe evauate { ddz dzd z dd} whee S is the sufae of the ube,, z. 66. B tasfoig to a tie itega evauate, ddz S S zdd whee S is the osed sufae bouded b the aes =, =6 ad ide + =a. 67. A divegee theoe to evauate { z ddz z dzd dd} whee S is the sufae of the shee + +z =. S 7
68. Veif Stoke s theoe fo the veto fied F = - i + j itegated oud the etage i the ae z = ad bouded b the ies =, =, = a ad =b. 69. Veif Stoke s theoe fo the veto fied F = - i-z j zk ove the ue haf sufae of + +z =, bouded b its ojetio o the ae. 7. Veif Gauss s Divegee theoe fo the futio F = i + j + z k ove the idia egio bouded b + =9, z= ad z=. 8