[試題] 周青松 微積分甲下（2006春 期末考）

[Nzing (黑色神話)]

課程名稱︰微積分甲
課程性質︰共同必修
課程教師︰周青松
開課系所︰
考試時間︰2006.06.19 13:25~15:05

試題 :

Ⅰ.A) For vectors a = a_1 i + a_2 j + a_3 k, b = b_1 i + b_2 j + b_3 k, and
      c = c_1 i + c_2 j + c_3 k, show that

                                |  a_1  a_2  a_3  |
                (a ╳ b) ‧ c = |  b_1  b_2  b_3  |.                    (15%)
                                |  c_1  c_2  c_3  |

   B) Let γ be a differentiable vector function of t and set r = ║γ║.
      Show that where r > 0

                   d    γ      1            dγ
                 ── (──) = ── [(γ ╳ ──) ╳ γ].               (15%)
                  dt    r      r^3           dt


                                                  2            2
Ⅱ. Find the directional derivative of f(x,y) = Ax  + 2Bxy + Cy  at (a,b)
    toward (b,a),
    A) if a > b;
    B) if a < b.                                                        (20%)


Ⅲ. Set r = ║γ║ where γ = xi + yj + zk. If f is a continuous
    differentiable function of r, then
                                             γ
                           ▽[f(r)] = f'(r) ──.                       (20%)
                                             r


Ⅳ. Evaluate the double integral

             3
    ∫∫ (3xy  - y) dxdy, Ω is the region between y = |x| and y = -|x|,
     Ω
    x in [-1,1].                                                        (15%)


Ⅴ. Evalutate the triple integral

              x
    ∫∫∫ 2ye  dxdydz, where T is the solial given by 0 ≦ y ≦ 1,
      T
    0 ≦ x ≦ y, 0 ≦ z ≦ x + y.                                       (15%)

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