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）︰100.4.16 考試時限（分鐘）：110分鐘 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. Close book. Please details of yours calculation.Good luck! → → 1.The vector field F is defined by F=(xz)j. Let S be portion of the surface → z=4-y^2 cut off by the planes x=0,z=0,and y=x let n be the normal vector field on S which points away forom the orgin. → → (a)(10%)Calculate the circulation over S using ∫∫(▽×F)‧n dσ directly s (b)(10%)Calculate the circulation over S by using Stoke's Theorem. 2.(10%)Let v be the volume encolsed by the surface S. u is a scalar feild du Please derive that ∫[▽u‧▽u+ u▽^2 u ]dV=∫ u ─dσ v s dn → → → 3.(10%) F(x,y)=(e^-y -2x)i-(xe^-y +siny)j is conservative. Cis the first → quadrant of the circle. R(t)=πcos(t)-πsin(t) for 0≦t≦π/2. Find the → → → potential of F and evalcuate the line integral ∫F‧dR . 4. ╭ 0 for -π≦x≦0 Let f(x)= ┤ ╰ xsinx for 0≦x≦π (a).(10%)Find tje Fourier series of f(x) on [-π,π] (b).(5%)Following (a),determine what this series converges to at x=11, x=10.5π, x=101π ,respectively. (c).(5%)Suppose f(x)is periodic with fundamental period 2π,write the Fourier series of f(x) in phase angle form. (d).(5%)Followinf (c).draw the amplitude spectrum of f(x) ^ 5. (a)(10%)Given F{f(t)}(ω)=f(ω), please derive that 1 ^ F{f(at)}(ω)= ──f(ω) for non-zero a. ｜a｜ ╭ 4-t^2 for -2≦t≦2 (b)(8%)Find the Fourier transform of f(t) ┤ ╰ 0 for (t<-2)∩(t>2) ^ sin(ω)-ωcos(ω) (c)(7%)Find the inverse Fourier transform of f(ω)= ───────── ω^3 (Hint:You may apply the result from(b)) (d)(10%)Following (c),find the inverse Fourier sin(ω)=ωcos(ω) 1 transform of ────────── (Hint:F{H(t)e^-at}= ─────) ω^3 a+ιω --

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