92 çB$ø`ç‚5 (-ç‚)
1.(15%) Ê7™(ρ, φ, θ) 5-, 7Þρ = aDÆVÞφ = π6FˇAíä , °w”
-í™(x, y, z)
2.(10%) Find the area of the part of the sphere x2+ y2+ z2 = a2 that lies within the cylinder x2+ y2 = ax and above the xy-plane
3.(10%) ° }RR
Dsin(9x2+ 4y2)dA, w2 D Ñ9x2 + 4y2 ≤ 15ä
4.(10%) q(C : ~r(t) = cos t ~i + sin t ~j + t ~k, 0 ≤ t ≤ 6π, wò
Ñρ(x, y, z) = 1 + xz, °¤(픾
5.(10%) I²¾Ò ~F = (exsin xy+yexcos xy)~i+(exy cos xy+z)~j +(zez+y)~k, C Ñâ(0, 2, 1)ƒ(1,π2, 2)5L<Ë(
[a] ° ~F íP?ƒb f (potential function), ¹∇f = F
[b] °R
CF · ~~ T ds 5M
6.(10%) Let ~a be a constant vector. ~r = x ~i + y ~j + z ~k is the position vector of (x, y, z), r = |~r|
Define the vector field ~V (x, y, z) = rn~a ×~r,where n is a positive integer.
Find div(~V ) and curl(~V )
7.(10%) Find the work done by the force ~F (x, y) = (x(x + y))~i + (xy2)~j in moving a particle from the origin along the x-axis to (1, 0), then along the line segment to (0, 1), and then back to the origin along the y-axis
8.(15%) Let (a, b, c) be a fixed point on the sphere S : x2+y2+z2 = R2(R >
0 is the positive radius). The mass density ρ(x, y, z) at (x, y, z) on S is
the distance from (x, y, z) to (a, b, c)(i.e. ρ(x, y, z)=p(x − a)2 + (y − b)2+ (z − c)2).
Find the total mass of S
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9.(15%) Find the outward flux of the vector field
V (x, y, z) = ((x/r~ 3) + y + z) ~i + ((y/r3) + x + z) ~j + ((z/r3) + x + y) ~k across the boundary of the ellipsoid region D : 10x2+ 11y2+ 12z2 ≤ 13, where r =px2+ y2+ z2
10.(15%) Define the vector field on the plane by
V (x, y) =~ −y ~i + (x2+ y2− x) ~j (x − 1)2+ y2 Prove that
[a](5%) curl(~V ) = 0
[b](10%) Compute the line integral H
ΓV · d~r, where Γ is a simple closed curve without passing (1, 0)
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