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f
Γ
f(x, y) = f(y, x) Γ y = x
x = y ′ y = x ′ f(x, y) = f(y ′ , x ′ ) = f(x ′ , y ′ ) (x, y) f
(x ′ , y ′ ) f
f(x, y) = −f(y, x)
f(x, y) = f(x, −y) Γ
x = x ′ y = −y ′ f(x, y) = f(x ′ , −y ′ ) = f(x ′ , y ′ ) (x, y)
f (x ′ , y ′ ) f
f(x, y) = −f(x, −y)
f(x, y) = f(−x, y) Γ
x = −x ′ y = y ′ f(x, y) = f(−x ′ , y ′ ) = f(x ′ , y ′ ) (x, y)
f (x ′ , y ′ )
f f(x, y) = −f(−x, y)
f(x, y) = f(−x, −y) Γ
x = −x ′ y = −y ′ f(x, y) = f(−x ′ , −y ′ ) = f(x ′ , y ′ ) (x, y)
f (x ′ , y ′ ) f
f(x, y) = −f(−x, −y)
g(ρ, θ) = 0
ρ ≥ 0 g Γ
0 < α < 2π g(ρ, θ + α) = ±g(ρ, θ) Γ
nα n ∈ Z
y = mx
f(x, mx) = 0 x = k(m) m
x = k(m) y = mk(m)
m∗ ∈ R lim k(m) = ±∞
m→m∗ y = m∗x + q
q = lim
m→m∗ mk(m) − m∗k(m) ∈ R
(x0, y0) ∈ R2 Γ
f(x0, y0) = 0
df(x0, y0) = ∂xf(x0, y0) dx + ∂yf(x0, y0) dy.
Γ (x0, y0)
∂xf(x0, y0)x + ∂y(x0, y0)y = q q
(x0, y0) q = ∂xf(x0, y0)x0 +∂y(x0, y0)y0
∇f(x0, y0) = (0, 0)
∇f(x0, y0) · (x − x0, y − y0) = 0,
(x0, y0) y−y0 = m(x−x0) x−x0 = m(y−y0)
y = ϕ(x) = mk(m) x = k(m)