課程名稱︰工程數學 課程性質︰大二必修 課程教師︰施文彬 開課學院：工學院 開課系所︰機械系 考試日期（年月日）︰2008.4.14 考試時限（分鐘）：110 mins 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : Rule : No calculators are allowed . You are allowed to bring an A4 size information sheet . Please provide 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 . (b)(10%)Let F = (2xy+z^2)i + (x^2+2yz)j +(2xz+y^2)k . Let C be the elliptic arc parameterized by r(t) = cos(t)i + cos(t)j + sin(t)k , 0 ≦ t ≦ π/2 Evaluate ∫F˙Tds by finding a potential function of F first . 2.(20%)Evaluate ∫∫F˙Ndσ , where F = (x^2)i - (e^z)j +(z)k and Σ is the surface bounding the cylinder x^2 + y^2 ≦ 4 , z ≦ 2 . (including the top and bottom caps of the cylinder) 3.(20%)Find the Fourier series of f(x) = cos(x) for -2≦x<0 and sin(x) for 0≦x≦2 . Also , determine what this series converges to for -2 ≦ x ≦ 2 . 4.(a)(5%)Please find the inverse Fourier transform of the function 6iω 12e ──── cos(2ω) . 16+ω^2 (b)(5%)Prove that Τ{H(t+a) - H(t-a)} = 2sin(aω)/ω . (c)(15%)Use Fourier Transform and Convolution to solve the differential equation y' + 2y = H(t+a) - H(t-a) .(Hint : Τ{H(t)e^(-at) = 1/a+iω} . 5.(15%)Let S be the portion of the sphere x^2 + y^2 + z^2 = 4 that lies below the plane z = 1 . Let n be the normal vector field on S which points away from the origin . -yz xz xyz Let F(x,y,z) = ─────i + ─────j - ─────k . x^2+y^2+1 x^2+y^2+1 x^2+y^2+1 Compute ∫∫(▽×F)˙ndσ . --

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