推 kenek:台語歌星?? 04/27 02:57
推 prinsces:......樓上 XD 05/15 02:22
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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