[試題] 100下 施文彬 工程數學下 期中考
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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04/20 21:45, , 1F
04/20 21:45, 1F