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F
Fc := {(x, y) ∈ A : F (x, y) = c}
N(x, y(x)) dy
dx
+ M(x, y(x)) = 0, M(x(y), y) + N(x(y), y) = 0.
dx dy
ω = 0 γ C1
P (x0, y0) (x, y) ∈ D ω 0 P
F (x, y) = ω
D
P (x0, y) (x, y) P (x, y0) (x, y)
F (x, y) =
F (x, y) =
x
x0
x
x0
γ
M(t, y) dt +
y
y0
y
M(t, y0) dt +
y0
N(x0, s) ds.
N(x, s) ds.
ω(x, y) C l ω G ω = 0
C l+1 D λ ∈ C l (A, R)
∂xG(x, y) = λ(x, y)p(x, y)
∂yG(x, y) = λ(x, y)q(x, y)
λ ∈ C l (D, R) λω λω
ω = 0
R 3
R 3
ω(x, y, z) = P (x, y, z) dx + Q(x, y, z) dy + R(x, y, z) dz
R 3 ω = 0
P (∂zQ − ∂yR) + Q(∂xR − ∂zP ) + R(∂yP − ∂xQ) = 0
ω(x, y, z) = dF (x, y, z) F (x, y, z) = C ∈ R
ω(x, y, z) λ(x, y, z)
λω = dF F (x, y, z) = C ∈ R λ(x, y, z) = 0
z
φ(z)
φ(z)
R 3
ω1(x, y, z) = 0 ω2(x, y, z) = 0 F1(x, y, z) = C1 ∈ R
F2(x, y, z) = C2 ∈ R
ω1 ω2 λ1 λ2
ω1 ω2 ω1 = 0 F1(x, y, z) = C
ω1 = 0 ω2 = 0