推 lovehan :純推施文彬 02/22 19:36
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）︰2007.4.16
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%) 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, as depicted below.
y│ .————— .
│ ／ ╱r(t) ＼
│ ∕ ╱ ﹨
│/ ╱ \
j↑╱ |
→─────────────
i x
→ → → →
(b)(10%) Let F =(2xy + z^2)i + (x^2 + 2yz)j + (2xz + y^2)k . Let C be the
→ → → →
elliptic are parameterized by r(t)= cos(t)i + cos(t)j + sin(t)k ,
→ →
0≦t≦π/2. Evaluate ∫ F‧T ds by finding a potential function
→ D
of F first.
→ →
2.(20%) Given the vector field F = yk and the surface Σ: x^2 - y^2 + z^2 =1,
→ → →
where 0≦y≦1 and z≧0. Please calculate ∫∫ F‧n dσ, where n is
Σ
the unit normal vector of Σ in the direction of increasing z.
→ → → →
3.(20%) Let F= zi + xj + yk and C be a circle contained in the plane x+y+z=1.
Show that the circulation of F around C anticlockwise when viewed
from (1,1,1) is √3 times the area of the disk enclosed by C.
4. Let f(x)= xsin(x) for -π≦x≦π.
(a)(10%) Write the Fourier series for f(x) on [-π,π].
(b)(10%) Obtain the Fourier expansion of sin(x)+xcos(x) on [-π,π].
5. f has period 3 and f(x)=┌ x for 0≦x＜2.
└ 0 for 2≦x＜3
(a)(8%) Find the complex Fourier series of f.
(b)(7%) Determine what this series converges to for -4≦x≦1.
(c)(5%) Plot the amplitude spectrum of f.
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※ 編輯: dakang 來自: 114.45.97.11 (02/11 02:46)