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ˆn = ∇G = ((∇G)1, (∇G)2, (∇G)3) ˆn = −∇G
⎛
⎞
∂φ1 ∂φ2 ∂φ3
⎜ ∂x ∂x ∂x ⎟ ⎛
⎞
⎜
⎟ 1 0 −2x
⎜
det ⎜ ∂φ1 ∂φ2 ∂φ3
⎟ = det ⎝ 0 1 −2y ⎠ = 4y
⎜ ∂y ∂y ∂y
⎟ 2x 2y 1
⎝
⎠
2 + 4x 2 > 0
(∇G)1 (∇G)2 (∇G)3
ˆn = ∇G(x, y) = (2x, 2y, 1) F
⎛
rot F = det ⎝ e1
e2
∂x
∂y
F1
F2
⎞ ⎛
⎠ = det ⎝ e1 ∂x y2 e2 ∂y 0
⎞
⎠ = (−1, −1, −2y)
e3 ∂z x − y
e3 ∂z F3
rot F · ˆn dσ = (−1, −1, −2y) · (2x, 2y, 1) dxdy = −
S
D
x = r cos θ y = r sin θ
1 π/2
1
rot F · ˆn dσ = − 2r(cos θ + 2 sin θ) r dθ dr = − 2r 2 dr ·
S
0
0
rot F · ˆn dσ =
S
+∂S
0
F dγ.
D
π/2
0
2x + 4y dxdy
(cos θ − 2 sin θ) dθ = 2
3 .
D γ1(θ) = (cos θ, sin θ) −π/2 <
θ < 0 γ2(t) = (1 − t, 0) 0 < t < 1 γ3(s) = (0, −s) 0 < s < 1 S
γ1 γ2 γ3 φ S
φ◦γ1
φ◦γ2
φ◦γ3
F dγ =
φ ◦ γ1(θ) = (cos θ, sin θ, 1 − cos 2 θ − sin 2 θ) = (cos θ, sin θ, 0);
φ ◦ γ2(t) = (1 − t, 0, 1 − (1 − t) 2 ) = (1 − t, 0, 2t − t 2 );
φ ◦ γ3(s) = (0, −s, 1 − s 2 ).
=
F dγ =
F dγ =
0
+∂S
−π/2
π/2
0
w=cos θ
= 1 −
1
0
1
0
F dγ =
φ◦γ1
F dγ + F dγ + F dγ
φ◦γ2
φ◦γ3
(sin 2 0
θ, 0, cos θ − sin θ) · (− sin θ, cos θ, 0) dθ = −
sin 3 θ dθ =
0
1
π/2
0
−w 2 dw = 2
3
(0, 0, 1 − t) · (−1, 0, 2 − 2t) dt =
(−s 2 , 0, s) · (0, −1, −2s) dt =
sin θ(1 − cos 2 θ) dθ = 1 −
1
0
1
0
π/2
0
2(1 − t) 2 dt = 2
3 .
−2s 2 dt = − 2
3 .
−π/2
sin 3 θ dθ
cos 2 θ sin θ dθ