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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