64 m kj izns'k jktf"kZ V.Mu eqDr fo'ofo|ky;] bykgkckn

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26 88 m kj izns’k jktf"kZ V.Mu eqDr fo’ofo|ky;] bykgkckn fo’k; % xf.kr Subject : Mathematics dkslZ “kh’kZd % Advance Calculus Courset Title : Advance Calculus vf/kU;kl (Assignment) 2012-2013 Lukrd dyk dk;ZØe Under Graduate Art Programme Section - A [k.M & d fo’k; dksM% ;w0th0,e0,e0 Subject Code : UGMM dkslZ dksM % ;w0th0,e0,e0-07 Course Code : UGMM-07 vf/kdre vad % 30 Maximum Marks:30 vf/kdre vad % 18 Max. Marks: 18 uksV % nhkZ mRrjh; iz”uA vius iz”uksa ds mRrj 800 ls 1000 “kCnksa esa fy[ksaA lHkh iz”u vfuok;Z gSA Note : Long Answer Questions. Answer should be given in 800 to 1000 words. Answer all questions. 1(a) (b) ( x, y, z) Calculate the Jacobian for 3 ( r, q, z) tSdksfc;u Kkr djsaA x = r cosq, y = r sinq , z = z 2 2 x x Prove that F = TR ® TR defined by f(x,y) = ( e cos y, e sin y) is not invertible on the whole of TR 2 , but it is locally invertible at each point of TR 2 . 3 2(a) Find ò ò D Kkr djsaA f ( x, y) dxdy where ìSiny f ( x, y) = í î y , y ¹ 0, y = 0 3 (b) Find the volume of the region lying in the first octant which is common to the two cylinders x 2 + y 2 = a 2 and x 2 + z 2 = a 2 3 ml ifj{ks= dk vk;ru Kkr djsa tks x 2 + y 2 = a 2 rFkk x 2 + z 2 = a 2 ds chp esa fLFkr gSA 3(a) Evaluate lim x + cos x (i) x ® -¥ x + sin x 3 Kkr djsa lim 1 (ii) x ® ¥ 1+ log( x - 2) (b) If A, B, C are the angles of a triangle such that 3 2 2 2 sin A + sin B + sin c = K dA tanC - tan B = dB tan A - tanC 2 2 2 ;fn A, B, C ,d f=Hkqt ds dks.k gS rFkk sin A + sin B + sin c = K gks dA tanC - tan B rks fn[kk;sa = dB tan A - tanC Section - B [k.M & [k vf/kdre vad % 12 Max. Marks: 12 uksV % ykq mRrjh; iz”uA vius iz”uksa ds mRrj 200 ls 300 “kCnksa esa fy[ksaA lHkh iz”u vfuok;Z gSA Note : Short Answer Questions. Answer should be given in 200 to 300 words. Answer all questions.
27 4. If ;fn y = x y b + x a then show that y = x y b + x a rks fn[kk;sa fd dy dx dy dx x y-1 y log y + yx x 1 y xy + x log x = - x y-1 y log y + yx x 1 y xy + x log x = - 2 2 1 2 2 5- Let f(x,y) = ( x + y ) ,"( x, y) ÎTR show that f is continuous at (0,0) but f x (0,0) & f y (0,0) do not exists. 3 2 5 6 Evaluate ò ( x ydx + 2x ydy) by aplying Greens theorem where 0 x c 1, Kkr djksaA 0 y 1. 3 7- Evaluate 3 Kkr djks 3
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Page 3 and 4: 3 5. Derive the wave equation for a
Page 5 and 6: nks ;qxfer dq.Mfy;ksa }kjk iw.kZ vo
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Page 19 and 20: 19 - ;fn y = (sin 1 x) gks rks fn[k
Page 21 and 22: 21 é1 2 3ù A = ê ú ê 0 5 0 3
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Page 74: 74 8- lgh fodYi dks fVd djsaA ¼d½
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81 efJe
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87 6. efJeÅeeLeea meneÙelee mesJe
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89 117 Gòej ØeosMe jepeef
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153 OTHER INFORMATION : Volume 2 :
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189 ╪ ≥ Γ ┼ ┘ 96 G≥ ej e
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64 m kj izns'k jktf"kZ V.Mu eqDr fo'ofo|ky;] bykgkckn

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