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台灣大學 106 年度 微積分(A)

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Vector calculus 2 題Infinite Series 1 題Applications of integration 1 題Parametric equations and polar coordinates 1 題

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第 1 題20 分Infinite Series
Determine the convergence or divergence of the following series:
∑n=1∞(cos⁡(πn)tanh⁡(πn)−1−cosh⁡(1n)sin⁡(1n)).\sum_{n=1}^{\infty} \left( \frac{\cos(\pi n)}{\tanh(\pi n)} - \frac{1 - \cosh(\frac{1}{n})}{\sin(\frac{1}{n})} \right).
第 2 題20 分Vector calculus
For a second continuously differentiable function, f(x,y,z)f(x, y, z), defined on the space R3\mathbb{R}^3, its Laplacian is defined to be
Δf=∂2f∂x2+∂2f∂y2+∂2f∂z2.\Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}.
Derive the formula of Δf\Delta f in the spherical coordinate system.
第 3 題20 分Applications of integration
Suppose that f(x)>0f(x) > 0 is a continuously differentiable function defined on x≥1x \geq 1. Let SS be the surface of revolution of the graph y=f(x)y = f(x) about the xx-axis. Let E⊂R3E \subset \mathbb{R}^3 be the solid enclosed by SS and x=1x = 1. More precisely,
E={(x,y,z)∈R3∣x≥1 and y2+z2≤f(x) for each x≥1}.E = \{(x, y, z) \in \mathbb{R}^3 \mid x \geq 1 \text{ and } \sqrt{y^2 + z^2} \leq f(x) \text{ for each } x \geq 1\}.
If the surface area of SS is finite, prove that the volume of EE must also be finite.
第 4 題20 分Parametric equations and polar coordinates
Let CC be the curve defined by {(x,y)∈R2∣x4+2x2y2+y4−2x(x2−y2)=0}\{(x, y) \in \mathbb{R}^2 \mid x^4 + 2x^2y^2 + y^4 - 2x(x^2 - y^2) = 0\}.
(a)10 分
Sketch the curve CC.
(b)10 分
Calculate the area of the region enclosed by CC.
第 5 題20 分Vector calculus
Let a,b,ca, b, c be three positive numbers with a>b>ca > b > c. Let Σ\Sigma be the surface defined by
(x−b)2a2+(y−c)2b2+z2c2=4.\frac{(x - b)^2}{a^2} + \frac{(y - c)^2}{b^2} + \frac{z^2}{c^2} = 4.
Note that Σ\Sigma encloses a bounded solid. Set n⃗\vec{n} to be the unit outer normal with respect to that solid. Consider the vector field
F⃗=(x(x2+y2+z2)32+cos⁡(yz),y(x2+y2+z2)32+y,z(x2+y2+z2)32+ez2).\vec{F} = \left( \frac{x}{(x^2 + y^2 + z^2)^{\frac{3}{2}}} + \cos(yz), \frac{y}{(x^2 + y^2 + z^2)^{\frac{3}{2}}} + y, \frac{z}{(x^2 + y^2 + z^2)^{\frac{3}{2}}} + e^{z^2} \right).
Calculate the flux integral ∫ΣF⃗⋅n⃗dS\int_{\Sigma} \vec{F} \cdot \vec{n} dS.