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X K E X a, b ∈ R f : E × [a, b] → K
u X x ∈ E t ∈ [a, b] ∂uf(x, t)
E × [a, b] ∂uf(x, t) E
∂uF (x) =
b
a
∂uf(x, t) dt
X E X I R f : E × I → Y Y
Φ : E × I × I → Y
Ψ(x, α, β) =
β
α
f(x, t) dt
Ψ
Ψ α, β
∂βΨ(x, α, β) = f(x, β), ∂αΨ(x, α, β) = −f(x, α);
X ≈ Kn ∂if(x, t) i = 1...n
Ψ(x, α, β) α, β
Ψ ′ n
β
(x, α, β)(h, △α, △β) = ∂jf(x, t) dt hj + f(x, β)△β − f(x, α)△α,
j=1
α
h = (h1, ..., hn) ∈ Kn x ↦→ α(x) x → β(x) R I α, β : E → I ⊆ R
∂kG(x) =
β(x)
α(x)
G(x) =
β(x)
α(x)
f(x, t) dt
∂kf(x, t) dt + f(x, β(x))∂kβ(x) − f(x, α(x))∂kα(x).
f : R 2 → R
f(x, y) = x 2 sin y
f(x, y) = |x|
f(x, y) = |xy|
f(x, y) = |x| + |y|
f(x, y) = |xy|
f(x, y) = sign(2 − x 2 − y 2 ) |2 − x 2 − y 2 |
∂xf(x, y) = 2x sin y ∂yf(x, y) = x 2 cos y R 2
R 2 Df(x, y) = 2x sin y dx + x 2 cos y dy.
∂xf(x, y) = sign(x)
2 |x| x = 0 ∂yf(x, y) = 0 R 2 \ ({0} × R)
Df(x, y) = sign(x)
2 |x| dx
∂xf(x, y) = |y| sign(x) ∂yf(x, y) = |x| sign(y)
xy = 0 Df(x, y) = |y|sign(x) dx + |x|sign(y) dy
∂xf(x, y) = sign(x) ∂yf(x, y) = sign(y) xy = 0
Df(x, y) = sign(x) dx + sign(y) dy