課程名稱︰高等微積分
課程性質︰系定必修
課程教師︰王振男 教授
開課系所︰數學系
考試時間︰2005/06/18 8:00-10:00
試題： Advanced Calculus Final (A)
1.(10%) If f:[0,∞) ×[0,1]→Ｒ is continuous, and
∞
F(y)＝∫ f(x,y) dx
0
converges uniformly on [0,1], prove that
1
∫ f(x,y) dy
0
is improperly integrable on [0,∞) and
1 ∞ ∞ 1
∫∫ f(x,y) dxdy ＝ ∫∫ f(x,y) dydx.
0 0 0 0
2. Compute the following integrals.
2
(a)(10%) ∫∫ e^(x+2y) dA where E is the trapezoid with vertices (1,1),
E
(2,2), (3,0), (6,0).
x^2 y^2
(b)(10%) ∫ xdy-ydx where C is the boundary of the ellipse ── + ── = 1
C a^2 b^2
oriented in the counterclockwise direction.
(c)(10%) ∫∫ F‧n dσ where S is the part of paraboloid z = 4－x^2－y^2
S
with z≧0, n is the upward-pointing normal, and F(x,y,z)＝(-x,y,3x^2).
2 2 2 2 2 2
(d)(10%) ∫∫ xy dydz + yz dzdx + x z dxdy where S is the sphere x +y +z =
S
4 with outward-pointing normal.
3. 3
(a)(10%) Show that if E屬於Ｒ satisfies the hypotheses of Gauss's Theorem,
then
∫∫∫(u△v＋▽u‧▽v)dV＝ ∫∫ u▽v‧n dσ
E E 的邊界
2
for all C functions u,v: E →Ｒ.
3
(b)(10%) Show that if E屬於Ｒ satisfies the hypotheses of Gauss's Theorem,
then
∫∫∫(u△v－v△u)dV＝ ∫∫ (u▽v－v▽u)‧n dσ
E E 的邊界
2
for all C functions u,v: E →Ｒ.
n
4.(15%) Let D be an open bounded domain in Ｒ with smooth boundary 邊界D
whose unit outer normal is denoted by n. Assume that u(x) is a nontrivial
2
C function satisfying
△u－u^3＝0 in D.
Show that there exists some x_0 屬於 邊界D such that
partial u
───── ≠ 0.
partial n
2
5. Let D＝{(x,y)屬於Ｒ : x^2＋y^2＜1} and B＝{(x,y,z)屬於Ｒ : x^2＋y^2＋z^2
2 _ 2 _ 2 _
＜1}. Suppose u屬於C (D\{0}), v屬於C (B\{0}) and w屬於C (B) satisfy
(1) lim u(x,y) = lim v(x,y,z) = ∞,
|x|+|y|→0 |x|+|y|+|z|→0
(2) u(x,y)＝0, for all (x,y)屬於邊界D,
(3) w(x,y,z)＝v(x,y,z)＝0, for all (x,y,z)屬於邊界B,
2 2 2
(4) (partial x＋partial y＋partial z) w(x,y,z)＝0, for all (x,y,z)屬於B.
Answer the following questions and prove or disprove your answers.
(i) Caculate the line integral ∮ ▽u‧T ds along the positive
邊界D
orientation, where T is the unit tangential vector. (5%)
(ii) Caculate the line integral ∮▽v‧T ds along the positive orientation,
3 Γ
where Γ＝{(x,y,0)屬於Ｒ : x^2＋y^2＝1}. (5%)
(iii) Can there exist a point (x_0,y_0,z_0)屬於B such that w(x_0,y_0,z_0)
＞0? (5%)
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