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I := arctan y dx − xy dy
γ
γ T (1, 0) (0, 1) (0, −1)
T
M := x 2 dx dy.
T
T = {(x, y) ∈ R 2 : −1 ≤ y ≤ 1, 0 ≤ x ≤ 1 − |y|}
γ = {(t, t − 1) : t ∈ [0, 1]} ∪ {(1 − t, t) : t ∈ [0, 1]} ∪ {(−t, 0) : t ∈ [1, −1]}
P (x, y) dx + Q(x, y) dy = arctan y dx − xy dy,
T
I = − (∂xQ − ∂yP ) dxdy = − y −
T
T
1
1 + y2
dx dy
1 1−|y|
= −
y − 1
1 + y2
dx dy
= −
= 2
−1
1
−1
1
0
0
y(1 − |y|) dy +
1 − y
1 + y 2
1
−1
1 − |y|
dy
1 + y2 = [2 arctan y − log(1 + y 2 )] y=1 π
y=0 =
M := x 2
dx dy =
Q(x, y) = x3 /3 P = 0
x
M = Q(x, y) dy =
γ
γ
3
dy =
3
= 1 1 1
+ =
12 12 6 .
M = x 2 dx dy =
= 1
3
= 2
3
T
T
1
−1
1
0
1
0
1
(1 − |y|) 3 dy
−1
T
t 3
3
(1 − y) 3 dy = 1
6 .
− log 2.
2
(∂xQ − ∂yP ) dxdy,
1 (1 − t)
dt +
0
3 1
dt − 0 dt
3
−1
1−|y|
x 2
dx dy
Γ = {(x, y) ∈ R 2 : (x 2 + y 2 + 12x + 9) 2 = 4(2x + 3) 3 },
0