x2+ y2 = 2 JLv‡j ²âõ

(1) °R R

Esin(x + y) cos(2x − y)dA, ¤T E uâ y = 2x − 1, y = 2x + 3, y = −x

¸ y = −x + 1FˇAí–. (Tý: ªà‰b‰² u = 2x − y, v = x + y) (12 })

Sol. Let u = 2x − y, v = x + y, then

y = 2x − 1 ⇒ u = 1 y = 2x + 3 ⇒ u = −3 y = −x ⇒ v = 0 y = −x + 1 ⇒ v = 1 Z Z

Exy

sin(x + y) cos(2x − y)dA = Z Z

Euv

sin v cos u · |∂(x, y)

∂(u, v)|dudv From u = 2x − y, v = x + y, we have 3x = u + v ⇒ x = ¹₃(u + v), y = ¹₃(−u + 2v). Thus, |^∂(x,y)_∂(u,v)| = ∂x/∂u ∂x/∂v

∂y/∂u ∂y/∂v =

1 3

1 3

−¹₃ ²₃ = ¹₃. The integral becomes

1 3

Z 1 0

Z 1

−3

sin v cos ududv = 1

3(sin 1 + sin 3)(1 − cos 1)

(2) Ê‰Ò ~F (x, y) = −y²~i + x²~j íTà-, •OÆ$˜ x²+ y² = 2 JLv‡j

²âõ (√

2, 0) Bõ (−√

2, 0) øÓñFÛdíŠÑÖý? (10 }) Sol. R F ·d~~ r =Rπ

0(−2 sin²θ, 2 cos²θ)·(−√

2 sin θ,√

2 cos θ)dθ = 2√ 2Rπ

0(cos³θ+

sin³θ)dθ Z π

0

cos³θdθ = Z π

0

(1 − sin²θ)d sin θ = sin θ −sin³θ 3 |^π₀

= 0

Z π 0

sin³θdθ = − Z π

0

(1 − cos²θ)d cos θ = −(cos θ −cos³θ 3 )|^π₀

= −(−1 + 1

3) + (1 −1 3)

= 2(1 −1 3) = 4

3

⇒RF · d~~ r = 2√

2 ·⁴₃ =⁸

√2 3

(3) t°( }

Z

C1S

C2

xydx + yzdy + zxdz,

¤T C1[ýâõ (0, 0, 0) Bõ (1, 1, 0) í²(¨, 7 C2[ýâõ (1, 1, 0) B õ (1, 1, 1) í²(¨. (12 })

1

Sol.

1. C1 = {ti + tj|0 ≤ t ≤ 1}. Take x = t, y = t, z = 0, dx = dt, then R

C1xydx + yzdy + xzdz =R1

0 t²dt = ¹₃

2. C2 = {i + j + tk|0 ≤ t ≤ 1}. Take x = 1, y = 1, z = t, dx = dy = 0, dz = 1, thenR

C2xydx + yzdy + xzdz =R1 0 tdt = ¹₂ 3. R

C₁S_C

2xydx + yzdy + zxdz =R

C₁xydx + yzdy + zxdz +R

C₂xydx + yzdy + zxdz = ⁵₆

(4) t°²¾Ò ~F íP‘ƒb ( potential function ), F =~ y

1 + x²y²~i + ( x

1 + x²y² + z

p1 − y²z²)~j + ( y

p1 − y²z² +1 z)~k (10 })

(5) t°²¾Ò ~F = (3xy −_1+y^x₂)~i + (e^x+ tan⁻¹y)~j â-D( r = 3(1 + cos θ)

²Õ¼|,¾ (outward flux). (12 })

Sol. M = 3xy − _1+y^x2, N = e^x+ tan⁻¹y ⇒ ^∂M_∂x = 3y − _1+y¹2, ^∂N_∂y = _1+y¹2 ⇒ Flux=R R

R(3y−_1+y¹₂+_1+y¹₂)dxdy =R R

R3y dxdy =R2π 0

Ra(1+cos θ)

0 (3r sin θ)rdrdθ = R2π

0 a³(1 + cos θ)³(sin θ)dθ = [−^a₄³(1 + cos θ)⁴]^2π₀ = −4a³− (−4a³) = 0 (6) °R R

SF · ~~ ndσ , w2 ~F = (x²+ y²)~k , S ÑÞ (z + xy)³= x²+ y² Ì„Ê 1 ≤ x²+ y²≤ 4 , ~n Ñ%,5¶²¾ (¹ ~n • ~k 5}¾ ≥ 0). (12 })

Sol. SÑf (x, y, z) = (z + xy)³− (x²+ y²)5PÞ.

∇f = (3(z + xy)²y − 2x, 3(z + xy)²x − 2y, 3(z + xy)²), 3(z + xy)² = 3(x² + y²)²³ ≥ 0. ~ndσ = ^∇f

|∇f ·~k|dxdy = ^∇f

3(x²+y²)²3

dxdy.

F · ~~ ndσ = (x²+ y²) · 3(x²+ y²)²³ · ^dxdy

3(x²+y²)²3

= (x²+ y²)dxdy = r³drdθ R R

SF · ~~ ndσ =R2π 0 dθR2

1 r³dr = 2π ·⁽²⁴₄⁻¹⁾ (7) IÞ S [ý7Þ x²+ y²+ z²= 4 \VÞ z =p

x²+ y²Fií¶}. t°Þ }R R

Sy²zdσ . (12 }) Sol. IR : x²+y²≤ 2, z =p

4 − x²− y²= f (x, y), dσ =q

1 + (^∂f_∂x)²+ (^∂f_∂y)²dxdy =

√ 2

4−x²−y²dxdy, Ix = r cos θ, y = r sin θ, 0 ≤ r ≤√

2, 0 ≤ θ ≤ 2π. FJ Z Z

S

y²zdσ = Z Z

S

y²p

4 − x²− y²· 2

p4 − x²− y²dxdy

= 2 Z Z

S

y²dxdy = 2 Z

√ 2

0

Z 2π 0

r²cos²θrdθdr

= 2 Z

√ 2

0

r³dr Z 2π

0

cos²θdθ = 2π

2

(8) q S u6Þ x²+ y² = 1, 0 ≤ z ≤ 1 , y‹,ÝQ x²+ y² ≤ 1,z = 1 F$A;

/q ~F = −y~i + x~j + x²~k. t°R R

S∇ × ~F · ~n dσ 5M. (10 }) Sol.

jø àStoke’s ìÜ, CÑx² + y² = 1, z = 0, r = cos ti + sin tj, r⁰ =

− sin ti + cos tj ⇒R R

S∇ × F · ndσ =H

CF · dr =R2π

0 1dt = 2π.

jù JòQlªœ,, ¤vI6Þ¶}S1, Õ¶²¾n1, ÝÞ¶}S2, Õ¶n2. ª)R R

S₁∇ × F · n1dσ = 0 (ÄÑ∇ × F · n1=-2xy). ∇ × F · n2= 2.

R R

S₂∇ × F · n2dσ =R R

S₂2dσ = 2π

(9) qÞSÑx²+ y²+ z²= 1, ~nѲÕÀP¶²¾. ~F (x, y, z) = 3xy²~i + 3x²y~j + z³~k. °Þ }R R

SF · ~~ ndσ. (12 }) Sol. ‚à Gauss ìÜ: IBÑx²+ y²+ z²≤ 1

Z Z

S

F · ~~ ndσ = Z Z Z

B

(∇ · ~F )dv

∇ · ~F = ∂

∂x(3x²y) + ∂

∂y3xy²+ ∂

∂z(z³)

= 3y²+ 3x²+ 3z²= 3(x²+ y²+ z²) Z Z Z

B

(∇ · ~F )dv = 3 Z Z Z

B

(x²+ y²+ z²)dxdydz

ùp7è™







x = ρ sin φ cos θ y = ρ sin φ sin θ z = ρ cos φ

= 3 Z 1

0

Z π 0

Z 2π 0

ρ²· ρ²sin φdθdφdρ

= 3 Z 1

0

ρ⁴dρ Z 2π

0

Z π 0

sin φdφdθ = 12 5 π

3