MAT 203 Vector Calculus [Kalkulus Vektor]
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1 UNIVERSITI SAINS MALAYSIA Fist Semeste Examination 0/0 Academic Session Januay 0 MAT 03 Vecto alculus [Kalkulus Vekto] uation : 3 hous [Masa : 3 jam] Please check that this examination pape consists of SIX pages of pinted mateial befoe you begin the examination. [Sila pastikan bahawa ketas pepeiksaan ini mengandungi ENAM muka suat yang becetak sebelum anda memulakan pepeiksaan ini.] Instuctions: Answe all five [5] questions. [Aahan: Jawab semua lima [5] soalan.] In the event of any discepancies, the English vesion shall be used. [Sekianya tedapat sebaang pecanggahan pada soalan pepeiksaan, vesi Bahasa Inggeis hendaklah diguna pakai]. /-
2 - - [MAT 03]. (a Find an equation of the plane in B(,,0 and (,3, 4 3 that passes though the points A(,,, [7 maks] (b Find the paametic equation fo the line though point B and pependicula to the plane in pat (a. [4 maks] (c A second plane passes though point (, 0, 4 and has nomal vecto, 4, 3. Show that this plane and the plane (a intesect at the point (, 0, 4 and the acute angle between them is appoximately [8 maks]. (a apatkan pesamaan satah dalam B(,,0 dan (, 3, 4. 3 yang melalui titik-titik A(,,, [7 makah] (b apatkan pesamaan paametik untuk gais luus yang melalui titik B dan seenjang kepada satah di bahagian (a. [4 makah] (c Suatu satah kedua melalui titik (, 0, 4 dan mempunyai vekto nomal, 4, 3. Tunjukkan bahawa satah ini dan satah (a besilang pada titik (, 0, 4 dan anggaan sudut di antaa meeka ialah [8 makah]. (a A paticle stats moving at the oigin with initial velocity i j + 3k. Its acceleation is time t. ( t= 6t t 6t (b The position vecto of an object in a i j k. Find the paticle position function at the 3 is ( t= ( acos t i ( asin t j + tk, a constant [8 maks] Show that if sum of the objects tangential and nomal component of acceleation a equal half its speed, then the angle. 4a [8 maks]...3/-
3 - 3 - [MAT 03] (c Let (t be a smooth vecto function in 3 and T is the unit tangent vecto with popety T. Show that κ T + N and hence show that κ on cuve. 3 whee κ is the cuvatue of the cuve descibed by (t at any point [6 maks]. (a Suatu zaah mula begeak dai asalan dengan halaju awal i j + 3k. Kecepatannya ialah ini pada masa t. ( t= 6t t 6t a i j k. apatkan fungsi kedudukan zaah [8 makah] (b Vekto kedudukan suatu objek dalam 3 ialah ( t= ( a cos t i ( a sin t j + tk, a pemala Tunjukkan bahawa jika hasil tambah komponen tangen kecepatan dan komponen nomal kecepatan besamaan sepauh lajunya, maka sudut a 4a (c Katakan (t suatu fungsi vekto yang licin dalam [8 makah] 3 dan T ialah vekto unit tangendengan dengan sifat T. Tunjukkan bahawa T + κ N dan seteusnya tunjukkan bahawa κ mana κ ialah kelengkungan lengkung yang diteangkan oleh sebaang titik di atas lengkung. 3 yang (t pada [6 makah] 3. (a Find the angle of intesection between the helix ( t = cos t i sin t j t k and the cuve 3 ( t = (+ t i t j t k at the point (, 0, 0. [6 maks] (b Let f ( x, y x xy y y. Find the diection u and the value of the deivative in the diection of u, u f fo the following,...4/-
4 - 4 - [MAT 03] (i u f (, is lagest. (ii u f (, is smallest. (iii u f (, 3 (c Show that the ellipsoid [9 maks] 3x y z 9 and the sphee x y z 8x 6y 8z 4 0 ae tangent to each othe at the point (,,. [8 maks] 3. (a apatkan sudut pesilangan di antaa helik lengkung ( t = cos t i sin t j t k dan 3 ( t = (+ t i t j t k pada titik (, 0, 0. [6 makah] (b Katakan f ( x, y x xy y y. apatkan aah u and nilai tebitan pada aah u, u f untuk beikut, (i u f (, tebesa. (ii u f (, tekecil. (iii u f (, 3 [9 makah] (c Tunjukkan bahawa elipsoid 3x y z 9 dan sfea x y z 8x 6y 8z 4 0 adalah tangen antaa satu sama lain pada titik (,, [8 makah] 4. (a Let be the cicle x y a, a 0. Show that at each point of cuve, yx, the vecto field F ( x, y in is paallel to the line tangent to at x y that point. [6 maks] (b Use the ivegence Theoem to find the outwad flux of 3 F= (6x xy i ( y x z j (4x y k acoss the bounday of the egion cut fom the fist octant by cylinde x y 4 and the plane z 3. [9 maks]...5/-
5 - 5 - [MAT 03] 4. (a Katakan suatu bulatan x y a, a 0. Tunjukkan bahawa pada setiap yx, titik di lengkung, medan vecto F ( x, y dalam adalah selai x y dengan gais tangen kepada pada titik itu. [6 makah] (b Gunakan Teoem ivegen untuk mendapatkan fluks kelua bagi 3 F= (6x xy i ( y x z j (4x y k meentasi sempadan kawasan yang dipotong oleh selinde x y 4 dan satah z 3 dalam oktan petama. [9 makah] 5. (a Given a vecto foms of Geen s Theoem F n ds div F( x, y da, is a positively oiented, piecewise-smooth and simple close cuve in a plane, is the egion bounded by and n is the outwad unit nomal vecto on the cuve. Pove that, f g da f ( g n ds f g da [7 maks] (b Evaluate x 4 xe dx ( x x y dy whee is the bounday of semi-annula egion in the uppe half-plane between cicle x y 4 and x y. [7 maks] (c Use Stoke Theoem to evaluate the flux of the cul of the field F= z i 3x j 5y k acoss the suface S : (, ( cos i ( sin j (4 k, 0 and 0 in the diection of the outwad unit nomal n, [7 maks]...6/-
6 - 6 - [MAT 03] 5. (a ibeikan suatu vekto bebentuk Geen s Theoem F n ds div F( x, y da, ialah suatu lengkung beoentasi positif, licin-cebis demi cebis dan tetutup mudah pada suatu satah, ialah kawasan yang dibatasi oleh dan n ialah vekto unit nomal atas lengkung. Buktikan bahawa, f g da f ( g n ds f g da [7 makah] (b Nilaikan x 4 xe dx ( x x y dy yang mana ialah sempadan bagi kawasan semiannula di antaa bulatan x y 4 dan x y pada sepauh bahagian atas satah. [7 makah] (c Gunakan Teoem Stoke untuk menilaikan fluk lingkaan suatu medan F= z i 3x j 5y k yang meentasi pemukaan S : (, ( cos i ( sin j (4 k, 0 dan 0 pada aah unit nomal n kelua. [7 makah] - ooo O ooo -
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