課程名稱︰工程數學
課程性質︰系必修
課程教師︰林沛群
開課學院：工學院
開課系所︰機械系
考試日期（年月日）︰2008.03.17
考試時限（分鐘）：120
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.(10%) Find the streamlines of the vector field F = i - y^2 j +z k ,
and then find the particular streamline through the given point (2,1,1).
2.(15%) A scalar field is defined as φ(x,y,z)= x^2 - y^2 - z^2.
Find the equation of normal line to the level surface φ(x,y,z)=0 at the
point Po = (1,1,0) ; then find the directional derivative of this scalar
field φ(x,y,z) at Po in the direction of the unit vector which lies in
the tangent plane of the level surface φ(x,y,z) = 0.
3.(10%) Find the mass and center of mass of a thin, straight wire from the
origin to (3,3,3) if δ(x,y,x) = x + y + z grams per centimeter.
4.(10%) Let C be a simple closed positively oriented path in the plane.
Let D consists of all points on C and in its interior. Let f,g,(pg/px),
(pf/py) (p表示偏微分符號) be continuous on D. Prove the second
half of the Green's Theorem: ∮C g(x,y)dy = ∫∫D (pg/px)dA.
5.(10%) f(x,y,z) = x^2 ,Σ is the part of the paraboloid z = 4- x^2 - y^2
lying above the x,y plane. Evaluate ∫∫Σ f(x,y,z)dσ .
6.(10%) Let Σ be a smooth closed surface bounding an interior M, show that
Volum of M = (1/3)∫∫Σ R‧N dσ
where R = x i + y j + z k is a position vector for Σ.
7.(15%)
Let F(x,y,z) = (yze^(xyz)-4x) i +(xze^(xyz)+z+cos(y)) j + (xye^(xyz)+y) k.
Test to see if F is conservative. If it is ,find a potential function
φ(x,y,z).
8.(20%) A five-turn circular spring with radius r = 1 and natural length
lo = 5 is shown in the figure . (圖太難畫了orz)
Assume the position vector of this spring is described as
R(u)= f(u) i + g(u) j + h(u) k .
(a) (5%) Assuming h(u) is parametrized by h(u)= u/(2π), obtain the complete
parametrized position vector R(u).
(b) (10%) Following (a), if the length of the spring is compressed to l = 2.5
,which one of the following has less percentage change, curvature or
torsion?
percentage change = abs((new-orig/orig))*100%
(c) (5%) A particle which starts at point Po (t=0) with zero speed is moving
along the trajectory of this spring (at its natural length). Assuming
the particle arrives at point P1 in one second (t=1) and during the
motion the speed increase linearly, find the position vector of this
particle R(t) which is parametrized by the variable "time" t.
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