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。 --

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