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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