作者ja000123 (懿轩)
站内NTU-Exam
标题[试题] 96下 周青松 微积分甲下 期末考
时间Mon Jun 30 17:54:43 2008
课程名称︰微积分甲
课程性质︰数学 - 微积分
课程教师︰周青松
开课学院:(如下)
开课系所︰生机、生工、地质、地理、工管等
考试日期(年月日)︰ 2008/06/13
黑色星期五
考试时限(分钟):8:10---10:00 迟到20分钟不得进场
是否需发放奖励金:是
(如未明确表示,则不予发放)
试题 :
I. For vector a=a1 i + a2 j + a3 k
and b=b1 i + b2 j + b3 k
A) Show that
| i j k|
a x b = |a1 a2 a3|
|b1 b2 b3|
2 2 2 2
B) Verify : ∥a x b∥ + (a · b) = ∥a∥∥b∥
II.
2
A) Find f(t) given that f'(t) = 2costi - tsint j + 2tk , and
f(0) = j + 3k
B) Let γ be a differentiable vector function of t and set r = ║γ║.
Show that, where r ≠ 0
d γ 1 dγ
── (──) = ── [(γ x ──) x γ]
dt r r^3 dt
III.
2
A) Find the function with gradient F(x,y,z) = yzi+ (xz + 2yz)j+ (xy + y )k
B) Find the directional derivative of f(x,y,x) = z㏑(x/y) at (1,1,2)
toward the point (2,2,1).
IV.
A) Let U be an open connected set and let f be a differentiable function on U.
If ▽f(x) = 0 for all x in U, then f is constant on U.
1 3 3
B) Use the chain rule to find the derivative of f(x,y) =—(x + y )
3
with sespect to t along the ellipse γ(t) = acosti+bsintj
V.
A) Evaluate the double integral
3
∫∫ (3xy - y) dxdy ,Ω is the region between y = |x| and y = -|x|,
Ω
x in [-1,1].
B) Evaluate the trible integral
x
∫∫∫ 2ye dxdydz, where T is the solial
T
∕ given by 0 ≦ y ≦ 1 , 0 ≦ x ≦ y
﹨ and 0 ≦ z ≦ x + y.
---每大题均20分---
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