MAT 203 Vector Calculus [Kalkulus Vektor]
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1 UNIVERSITI SAINS MALAYSIA First Smstr Eamination 009/00 Acadmic Sssion Novmbr 009 MAT 0 Vctor Calculus [Kalkulus Vktor] Duration : hours [Masa : jam] Plas chck that this amination papr consists o SEVEN pags o printd matrial bor ou bgin th amination. [Sila pastikan bahawa krtas ppriksaan ini mngandungi TUJUH muka surat ang brctak sblum anda mmulakan ppriksaan ini.] Instructions: Answr all thr [] qustions. [Arahan: Jawab smua tiga [] soalan.] In th vnt o an discrpancis, th English vrsion shall b usd. [Skirana trdapat sbarang prcanggahan pada soalan ppriksaan, vrsi Bahasa Inggris hndaklah diguna pakai]. /-
2 [MAT 0]. Instruction: Writ th choic o our answrs in th am answr script (a) I u,,, v 6,, ind th componnt o u in th dirction o v. 0 (E) non o th abov Find th quation o plan through th point,, with normal vctor i j k. z 8 z z (E) non o th abov Find a vctor paralll to th lin o intrsction o th plans z, and z. 6,, 6,,, 6,,, (E) non o th abov. (d) Calculat th distanc btwn th two plans z and 0 z 8. 6 (E) non o th abov. () () (g) Find th ara o th triangl which has its vrtics at th points P,,, Q,, and R,,. '0 6 (E) non o th abov. Th domain o th unction, is D, : D, : D, : 0 D, : 0 (E) non o th abov. Th limit lim, 0,0 dos not ist (E) non o th abov..../-
3 [MAT 0]. Arahan: Tuliskan jawapan pilihan anda di dalam krtas jawapan ppriksaan. (a) Jika u,,, v 6,, dapatkan komponn u di dalam arah v. 0 Dapatkan prsamaan satah ang mlalui titik,, dngan vktor normal i j k. z 8 z z Dapatkan vktor ang slari drngan garis prtindanan dua satah z, dan z. 6,, 6,,, 6,,, (d) Dapatkan jarak diantara dua satah z dan 0 z 8. 6 () Cari luas sgitiga ang mmpunai vrtk pada titik-titik P,,, Q,, dan R,,. '0 6 () (g) Domain bagi ungsi, adalah D, : D, : D, : 0 D, : 0 had, 0,0 adalah tidak wujud.../-
4 [MAT 0] (h) Find th lngth L o t cos t,sin t,t rom t 0 t (E) non o th abov. (i) Find th work don b F along an path C rom th point P,, point Q,, (E) non o th abov. to th (j) Find i,, z zat 0,, i k i j i i k (E) non o th abov. (k) Comput E 9 ddwhr E is th llips (E) non o th abov. 0 9 (l) Evaluat th tripl intgral 8 (E) z dzdd non o th abov. (m) Lt z ln z. Comput 0 (E) non o th abov. z at, and z (n) Evaluat 0 d (E) non o th abov..../-
5 [MAT 0] (h) Dapatkan panjang L bagi t cos t,sin t,t dari t 0 shingga t. (i) Dapatkan krja ang dilakukan F dispanjang laluan C brmula dari titik P,, shingga titik Q,, (j) Dapatkan jika,, z z pada titik 0,, i k i j i i k (k) Dapatkan E 9. dd E adalah sbuah llips 0 9 (l) Nilaikan z dzdd (E) 9 tiada satu pun jawapan di atas. (m) Biarkan z ln z. Dapatkan z pada, 0 (n) Nilaikan 0 d /-
6 6 [MAT 0] (o) Find th dirctional drivativ o th dirction o v,., at th point,, in (E) non o th abov.. (a) Show that th unction, sin cos satisis th wav quation 0. Find th quation o th tangnt plan to th surac point,,. z 9at th Find all local maima and minima o,.. (a) I s is th ntir, plan, valuat th intgral I ds. s b transorming th intgral into polar co-ordinats. A co-ordinat sstm u, k, w is rlatd to Cartsian co-ordinats, b ukw uk w,, u k / (i) Find th scal actors hu, hk, hw. (ii) Find th volum lmnt in th u, k, w sstm. Vri stoks thorm or th givn surac S dind b z, z 0, orintd b upward normal, F zi zj k..../-
7 [MAT 0] (o) Dapatkan trbitan brarah bagi dalam arah v,., pada titik,, di. (a) Tunjukkan bahawa ungsi, sin cos mmnuhi prsamaan glombang 0. Dapatkan prsamaan satah tangn bagi prmukaan titik,,. z 9pada Dapatkan smua titik maksimum tmpatan dan minimum tmpatan bagi,.. (a) Jika s adalah satah, nilaikan kamiran ini kpada bntuk koordinat polar. I ds, dngan mnukar bntuk Sistm koordinat u, k, w adalah brkait dngan koordinat kartsan, dngan ukw uk w,, u k / (i) Cari aktor skalar hu, hk, hw. (ii) Cari lmn ktumpatan di dalam sistm u, k, w. s Tntusahkan thorm stoks bagi prmukaan S ang ditakrikan sbagai z, z 0, dngan orintasi normal ang brarah kluar, F zi zj k. - ooo O ooo -
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