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)=(