课程名称︰工程数学 课程性质︰系必修 课程教师︰林沛群 开课学院：工学院 开课系所︰机械系 考试日期（年月日）︰2008.03.17 考试时限（分钟）：120 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 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表示偏微分符号) 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 .　(图太难画了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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