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C = −B
∂yu(x, y) =
C/x
1 + y2 y
+
/x2 x2 Cx + y
=
+ y2 x2 + y2 ∂xu(x, y) = − C y/x2
+ B
1 + x2 1 + y2 −
/x2 C
1 + 1/x 2
− 1
x2
+
x
x2 x + By
=
+ y2 x2 + y2 γ(t) = (cos θ, sin θ) θ ∈ [0, 2π]
2π
2π
ω = (cos θ + B sin θ)(− sin θ) dθ + (−B cos θ + sin θ) cos θ dθ
γ
=
0
2π
0
− sin θ cos θ − B sin 2 θ − B cos 2
θ + sin θ cos θ dθ = −2πB
R 2 \ {(0, 0)}
1 ω
B = C = 0 ũ(x, y) = 1
2 log(x2 +y 2 ) R 2 \{(0, 0)}
α
y
ω = 2 + dx + 1 + α
x
x
dy
y
x > 0 y > 0 U(x, y) U(1, 1) = 2
ω =
y
dx +
x
0
α x
y dy + 2 dx + dy =: ω1 + df2(x, y)
f2(x, y) = 2x + y ω ω1 ω1
∂ y ∂
= α
∂y x ∂x
x
y
1
2 √ xy = √ α ∂
α
∂x
x
y
1
2 √ 1
= α
xy 2 √ xy ,
α = 1
α
y x
ω = dx +
x y dy + df2(x, y)
u(x, y) = f1(x, y)+f2(x, y)+c f1 ω1 c ∈ R
ω1 (1, 1)
f1(x, y) = ω1,
γ (x,y) (1, 1) (x, y) ω1 =
ωx 1 dx + ωy 1 dy
f1(x, y) =
x
ω
1
x y
1 (s, 1) ds +
1
γ (x,y)
ω y
1 (x, t) dt = 2(√ x − 1) + (2 √ x)( √ y − 1) = 2( √ xy − 1).
f2(1, 1) = 3 f1(1, 1) = 0 c = −1
U(x, y) = 2 √ xy + 2x + y − 3.