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2 Green’s theorem : R R

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Å Cɽ¸×ç´½ ˙çÍ 2000 ˙bç ( ø ) úŸ×5j

1 The vector field a = (z 2 + 2xy)i + (x 2 + 2yz)j + (y 2 + 2xz)k. Show that a is conservative field and find the line integral of R

a · dr along any line joining (1, 1, 1) and (1, 2, 2). (10 %) Ans: Yes, 11.

2 Green’s theorem : R R

A ( ∂Q ∂x ∂P ∂y )dA = H

C (P dx + Qdy)

Determine the following line integrals and area integrals as shown in Fig.1 (25 %) (a). 1 2 { H

C {−ydx + xdy}, (b). 1 2 H

C | r × ˆt | ds, (c). 1 2 H

C r · ˆ n ds, (d). R

A (∇ × a) · kdA, and (e). R

A dA, where C is the circular boundary of a unit circle, ˆt is the unit tangent vector, ˆ n is the unit normal vector, (ds) 2 = dr · dr, a = (−y, x) and A is the area of a unit circle.

Ans: (a). π, (b). π, (c). π, (d). 2π, (e). π.

3 Gauss’ theorem: R R R

V ∇ · a dV = R R

S a · dS

(a). Find the volume enclosed between a sphere of radius 1 centered on the origin, and a circular cone of half angle 60 degrees with its vertex at the origin as shown in Fig.2. (10 %) (b). Using Gauss’ theorem, find the k value (5 %)

Z

V

∇ · r dV = Z

S

r · dS = k Z

V

dV.

(c). Given

F = r

(r 2 + a 2 ) 3/2 ,

find ∇ · F. (5 %) Find the volume integral R

V ∇ · FdV (10 %) and surface integral R

S F · dS, (10 %) where V is the volume of sphere | r |=

3a and S is the surface vector on the volume V .

Ans: (a). π/3, (b). k = 3, (c). 3a

2

(r

2

+a

2

)

5/2

, 3 3π/2.

4 Stokes’ theorem: R

S (∇ × a) · dS = H a · dr

Given a = yi − xj + zk, S : x 2 + y 2 + z 2 = a 2 , z > 0 and C : x 2 + y 2 = a 2 , z = 0, verify Stokes’

theorem by determining R

S (∇ × a) · dS and H

a · dr. (10 %) Also, calculate R

S dS = H

C r × dr.

(10 %)

Ans: −2πa 2 . 2πk.

5 Please explain the relationship among Green’s, Stokes’ and Gauss’ theorems. (10 %) Ans: Green’s theorem can derived from Gauss’ and Stokes’ theorems, respectively.

½×´ Í 2001 úŸ×5 by Chen for vector calculus

æf:big03s.ctxf:Jan./12/2001 A3 (75 M)

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