→ randyko :第1題可是讓人算到吐血阿... 02/10 00:18
推 iamapigtoo :沒記錯的話，第一題曾經是作業... 02/10 00:52
→ iamapigtoo :呀，沒注意到是九四年的 02/10 00:54
※ 編輯: dakang 來自: 114.45.97.11 (02/11 02:46)
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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