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C R 2
x = ξ, y = ξη
1
N(1, η)
dξ +
dη = 0.
ξ M(1, η) + ηN(1, η)
M, N α = −1
D
F (x, y) = 1
[x · M(x, y) + y · N(x, y)].
α + 1
A ω = 0
g : A → R C1 gω ω = 0
gω = 0 g > 0 g = ef f : A → R
ω = p(x, y) dx + q(x, y) dy
ef ω
∂yp − ∂xq = −p∂yf + q∂xf
∂yp − ∂xq = h(x)q e h(x) dx
∂yp(x, y) − ∂xq(x, y)
q(x, y)
x e h(x) dx
h(x) = ∂yp(x, y) − ∂xq(x, y)
;
q(x, y)
∂yp − ∂xq = k(y)p e − k(y) dy
∂yp(x, y) − ∂xq(x, y)
−p(x, y)
y e k(y) dy
k(y) = ∂yp(x, y) − ∂xq(x, y)
;
−p(x, y)
∂yp − ∂xq = f(x)q(x, y) − g(y)p(x, y)
f, q, p C1
x y
h(x, y) = exp f(t)dt + g(t)dt
ω
x0
x r y s (my dx + nx dy) + x ρ y σ (µy dx + νx dy) = 0
r, s, ρ, σ, m, n, µ, ν mν − nµ = 0 x α y β
α, β
M(x, y) dx + N(x, y) dy = yf(xy) dx + xg(xy) dy = 0
1
f = g
Mx − Ny
C ⊆ R 2 R 2 (x, y) ∈ C (kx, ky) ∈ C
k > 0
y0