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）︰2011/04/28
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 :
A.Consider a circle C, which lies in the xz-plane, with center (a,0,0)
and radius r < |a|. Let Γ be the torus obtained by rotating C about z-axis.
Find the tangent plane of Γ at point
( (a/2^(1/2))+(r/2) , (a/2^(1/2))+(r/2) , (r/2^(1/2)) )
B.Calculate the first fundamental form of the surface of revolution given
by r = (x^2+y^2)^(1/2) = f(z), where f is a C-1 function, in terms of the
-1
cylindrinates z and θ=tan (y/x).
2 2 2
C.Let S be the sphere x + y + z = 1
(a) Use stereographic projection from the north pole (0,0,1) to the
plane z = 0 to obtain a parametric representation for S\{(0,0,1)}
2 3
(b) Show that the parametrization r(u,v):R -> R in (a) is conformal.
That is, if two curves on z = 0, which intersect at (u,v,0), then the
two image curves on the sphere are also orthogonal at r(u,v)
2 2
D.Consider the function U = F(X) = (x - y , xy).
(a) Obtain an iterative approximation G(X), which depends on given U,
-1
for the inverse transformation F (U) near X_0 = (1,1) or U_0 = (0,1).
Verify that the fixed point X_fixed of G satisfies U = F(X_fixed).
(b) Show that there exists a δ>0 s.t. for any U ∈ B_δ(U_0) the iteration
X_n+1 = G(X_n) with initial value X_0 converges to a limit,denoted by X(U).
註:
X_0 : 0是下標，其他同理。
--
※ 發信站: 批踢踢實業坊(ptt.cc)
◆ From: 140.112.240.100
※ 編輯: impin 來自: 140.112.240.100 (06/08 12:31)
※ 編輯: impin 來自: 140.112.240.100 (06/08 12:33)