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I R r : I → Rd
γ µ : r(I) → [0, +∞[
massa(γ) := µ(r(t)) |r ′ (t)| dt = 0,
r µ
r(t)µ(r(t)) |r
G I :=
′ (t)| dt
.
massa(γ)
d = 2 r(t) = (x(t), y(t))
⎛
⎜
x(t)µ(x(t), y(t))
G := ⎜ I
⎝
˙x 2 (t) + ˙y 2
(t) dt y(t)µ(x(t), y(t))
I ,
massa(γ)
˙x 2 (t) + ˙y 2 ⎞
(t) dt
⎟
massa(γ)
⎠ .
d = 2 µ = cost
massa(γ) = µ |r ′ (t)| dt = µ lunghezza(γ),
I
I
G
1
:=
x(t)
lunghezza(γ) I
˙x 2 (t) + ˙y 2
(t) dt, y(t)
I
˙x 2 (t) + ˙y 2
(t) dt .
γ
y ≥ 0 µ > 0
γ r(t) = (x(t), y(t)) = (cos t, sin t) θ ∈ I := [0, π]
µ lunghezza(γ) = π
x(t)
I
˙x 2 (t) + ˙y 2 π
(t) dt = cos t sin
0
2 t + cos2 π
t dt = cos t dt = 0.
0
y(t) ˙x 2 (t) + ˙y 2 π
(t) dt = sin t sin2 t + cos2 π
t dt = sin t dt = 2.
G = (0, 2/π)
I
0
γ r(t) = (t 2 , 2t)
t ∈ I := [0, 1] µ > 0
x(t) = t2 y(t) = 2t
γ
|r ′ 1
1
(t)| dt = (2t) 2 + 22 dt = 2 1 + t2 dt.
I
α ∈ R
α
t
+ t2 dt = t α + t2 −
2 dt
= t α + t2 α
− + t2 dt + α
0
√ α + t 2 = t α + t 2 −
0
0
dt
√ α + t 2
α + t 2 − α dt
√ α + t 2
√ α + t2 = −t + w t = w2 − α
2w dt = w2 + α
2w2
dt
√
α + t2 =
2w
w2 w
+ α
2
+ α 1
dw =
2w2 w = log |w| + c = log |t + α + t2 | + c,