課程名稱︰工程數學下
課程性質︰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）︰2011/3/23
考試時限（分鐘）：90分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1.(15%)Find the streamlines of the vector field F=(1/x) i +(exp^x) j- k,and then
find the particular streamline through the given point(2,0,4).
2.(10%)Following problem 1,is F conservative?If it is,find its potential
function φ(x,y,z).
3.(15%)A scalar field is defined as φ(x,y,z)=cos(y-x)-exp^z. Compute the
directional derivative of the function in the direction of i-j+2k, and then
find the equations of the normal line and the tangent plane to the level
surface φ(x,y,z)=0 at the point P=(1,1,0).
4.(10%)Let Σ be a smooth closed surface bounding an interior M, show that
Vloume of M=1/3*∫∫R．N dσ, where R=xi+yj+zk.
5.(15%)Analytically express the normal vector of a planar thin wire y=S(x). In
addition,express its center of mass with density functionδ(x,y) per unit
length and range x=a to x=b.
6.(15%)Let D be a simply connected domain in 3-space. Let F and ▽×F be
continuous on D. If F is conservative, proof that ▽×F=0 in D.
7.(20%)Roughly sketch curvature vs. t & torsion vs. t of the following five
different curves in 3-space, and explain your reasons.
C1:x=t,y=2t,z=3t
C2:x=cos(t),y=sin(t),z=0
C3:x=t*cos(t),y=t*sin(t),z=0
C4:x=cos(t),y=sin(t),z=t/(2π)
C5:x=t*cos(t),y=t*sin(t),z=3t
where 0<=t<=10π for all C
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