課程名稱︰工程數學下 課程性質︰機械系大二下必修 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）︰2006.4.17 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 the details of your calculation. Good luck! 1.(a)(10%) Let C be a simple closed path in the x,y plane, with interior D. Let φ(x,y) and Ψ(x,y) be continuous with continuous first and second partial derivatives on C and throughout D. d^2(Ψ) d^2(Ψ) Let ▽^2(Ψ) = ───── + ─────. d^2(x^2) d^2(y^2) Prove that ∫∫φ▽^2(Ψ) dA D dΨ dΨ = ∫-φ──dx + φ──dy -∫∫▽φ‧▽Ψ dA. C dy dx D (b)(10%) Under the condition of (a), show that ∫∫(φ▽^2(Ψ)-Ψ▽^2(φ))dA D dφ dΨ dΨ dφ = ∫[Ψ── -φ──]dx + [φ── -Ψ──]dy C dy dy dx dx 2. Let f be the periodic function with period 2 given on (-1,1) by f(t) =┌ 1 -1＜t＜0 └ -1 1≦t＜1 (a)(15%) Find the Fourier series of f. (b)(5%) What does the Fourier series of converges to when t=1/2? When t=100? → 2 → → → 3.(a)(10%) Given F =(sec(x) +ln(y)) i +(x/y + ze^y) j + e^y k. Calculate the → work done by F along the line segment from (π/4,1,0) to (0,1,1). (b)(10%) 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)‧n dσ. S 4.(10%) Let f(x)= xsin(x) for -π≦x≦π. Write the Fourier series for f(x) on [-π,π]. 5. Given the closed region cut from the first octant by the coordinate plane and sphere x^2 + y^2 + z^2 = 4. Determine the outward flux of the vector → → → → field F = xzi + yzj + z^2 k through the surface in two different ways: (a)(15%) by direct calculation of the fluxes over the bounding surfaces; (b)(15%) using the Divergence Theorem. --

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