課程名稱︰工程數學下 課程性質︰必修 課程教師︰施文彬 開課學院：工學院 開課系所︰機械工程學系 考試日期（年月日）︰104/3/30 考試時限（分鐘）：110分鐘 試題 : Prelim I, Engineering Mathematics II, Spring 2015 Time: 10:20 ~ 12:10, Mar. 30, 2015. Rule: Calculators are not allowed. Please provide details of your calculation. Good luck! 1. (20%) The two surfaces, y = e^x sin(2πz) + 2 and, intersect in a curve, find equations of the tangent line to the curve of intersection at point (0,2,1). 2. (20%) A vector field is given as F = (2x)i + (2y)j + (3z)k. S is the portion of the paraboloidz = 4 - x^2 - y^2 above the xy plane. Calculate flux of F across S by applying Divergence Theorem. 3. (20%) Find the mass of the portion of the sphere x^2 + y^2 + z^2 = 4 in the first octant if the area density at any point (x,y,z) on the surfac e is kz^2, where k is a constant. 4. (20%) A vector field is given as F = [z + ln(x^2 + 1)]i + [cos(y) - x^2]j + (3y^2 - e^z)k. C has the position vector R = cos(t)i + sin(t)j + k where 0 ≦ t ≦ 2π. (a) Calculate circulation of F on C without using Stoke's Theorem. (b) Calculate circulation of F on C by applying Stoke's Theorem. 1 2z 1 x 2 y 5. (20%) Consider the force field F(x,y)=(--- - ---)i-(--- + ---)j+(--- + ---)k. y x^2 z y^2 x z^2 (a) Show F to be conservative and find a potential function f(x,y,z) for F. (b) Find the work done by F along any path from (2,-1,1) to (4,2,-2). http://imgur.com/eCOZ1gp --

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