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标题[转录][试题] 周青松 微积分甲下（2006春 期末考）
时间Sat Jun 7 19:01:19 2008
※ [本文转录自 NTU-Exam 看板]
作者: Nzing (黑色神话) 看板: NTU-Exam
标题: [试题] 周青松 微积分甲下（2006春 期末考）
时间: Tue Jun 20 20:41:18 2006
课程名称︰微积分甲
课程性质︰共同必修
课程教师︰周青松
开课系所︰
考试时间︰2006.06.19 13:25~15:05
试题 :
Ⅰ.A) For vectors a = a_1 i + a_2 j + a_3 k, b = b_1 i + b_2 j + b_3 k, and
c = c_1 i + c_2 j + c_3 k, show that
| a_1 a_2 a_3 |
(a ╳ b) ‧ c = | b_1 b_2 b_3 |. (15%)
| c_1 c_2 c_3 |
B) Let γ be a differentiable vector function of t and set r = ║γ║.
Show that where r > 0
d γ 1 dγ
── (──) = ── [(γ ╳ ──) ╳ γ]. (15%)
dt r r^3 dt
2 2
Ⅱ. Find the directional derivative of f(x,y) = Ax + 2Bxy + Cy at (a,b)
toward (b,a),
A) if a > b;
B) if a < b. (20%)
Ⅲ. Set r = ║γ║ where γ = xi + yj + zk. If f is a continuous
differentiable function of r, then
γ
▽[f(r)] = f'(r) ──. (20%)
r
Ⅳ. Evaluate the double integral
3
∫∫ (3xy - y) dxdy, Ω is the region between y = |x| and y = -|x|,
Ω
x in [-1,1]. (15%)
Ⅴ. Evalutate the triple integral
x
∫∫∫ 2ye dxdydz, where T is the solial given by 0 ≦ y ≦ 1,
T
0 ≦ x ≦ y, 0 ≦ z ≦ x + y. (15%)
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※ 编辑: Nzing 来自: 140.112.240.198 (06/20 20:41)
※ chaochienyao:转录至看板 NTUBIME100HW 05/30 08:28
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