課程名稱︰微積分甲下
課程性質︰必修
課程教師︰王金龍
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2011/3/24
考試時限（分鐘）：40
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
2 2
A. Let u(x,y) : R → R be a C function. Express u + u in polar coordinate.
xx yy
2 1
B. Let f(x,y) : R → R be a C function.
d b b
Show that ─ ∫ f(x,y) dx = ∫ f (x,y) dx.
dy a a y
Hint. Every continuous function defined on a bounded and closed set
n
D ㄈ R is uniformly continuous.
(ㄈ：包含於)
3
C. Consider the line integral ∫ L, where L = Adx + Bdy + Cdz defined on R .
Γ
3
Prove that L is exact , that is, L = df for somr f on R , if and only if
the integral is independent to the path, which means that it only depends
on the end points of Γ.
(Note that we assume A, B, C are all continuous)
┌ x = cos t
D. (a) Evaluate ∫ zdx + xdy + ydz over the arc of the helix │
│ y = sin t
│
└ z = t
from (1,0,0) to (1,0,2π).
ydx + xdy 1
(b) Evaluate ∫ ─────── over the arc of y = sin ─
1 + (x^2)(y^2) x
from (1/2π,0) to (1/π,0).
Hint. Check if the differential form is exact.
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