课程名称︰工程数学下 课程性质︰机械系大二下必修 课程教师︰施文彬 开课学院：工学院 开课系所︰机械系 考试日期（年月日）︰2007.4.16 考试时限（分钟）：110 mins 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : Rule: No calculators are allowed. You are allowed to bring an A4 size information sheet. Please provide the details of your calculation. Good luck! 1.(a)(10%) Determine the area enclosed by the curve parameterized by → → → r(t) = (t-sin(t))i + (1-cos(t))j , 0≦t≦2π, and the x-axis, as depicted below. y│ .————— . │ ／ ╱r(t) ＼ │ ∕ ╱ ﹨ │/ ╱ \ j↑╱ | →───────────── i x → → → → (b)(10%) Let F =(2xy + z^2)i + (x^2 + 2yz)j + (2xz + y^2)k . Let C be the → → → → elliptic are parameterized by r(t)= cos(t)i + cos(t)j + sin(t)k , → → 0≦t≦π/2. Evaluate ∫ F‧T ds by finding a potential function → D of F first. → → 2.(20%) Given the vector field F = yk and the surface Σ: x^2 - y^2 + z^2 =1, → → → where 0≦y≦1 and z≧0. Please calculate ∫∫ F‧n dσ, where n is Σ the unit normal vector of Σ in the direction of increasing z. → → → → 3.(20%) Let F= zi + xj + yk and C be a circle contained in the plane x+y+z=1. Show that the circulation of F around C anticlockwise when viewed from (1,1,1) is √3 times the area of the disk enclosed by C. 4. Let f(x)= xsin(x) for -π≦x≦π. (a)(10%) Write the Fourier series for f(x) on [-π,π]. (b)(10%) Obtain the Fourier expansion of sin(x)+xcos(x) on [-π,π]. 5. f has period 3 and f(x)=┌ x for 0≦x＜2. └ 0 for 2≦x＜3 (a)(8%) Find the complex Fourier series of f. (b)(7%) Determine what this series converges to for -4≦x≦1. (c)(5%) Plot the amplitude spectrum of f. --

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