Sea
f
:
A
→
R
{\displaystyle f:A\to \mathbb {R} }
A
⊆
R
n
{\displaystyle A\subseteq \mathbb {R} ^{n}}
A
{\displaystyle A}
(
a
1
,
a
2
,
…
,
a
n
)
∈
A
{\displaystyle (a_{1},a_{2},\dots ,a_{n})\in A}
∂
n
f
∂
x
i
.
.
.
∂
x
j
(
a
1
,
…
,
a
n
)
=
∂
n
f
∂
x
j
.
.
.
∂
x
i
(
a
1
,
…
,
a
n
)
.
{\displaystyle {\frac {\partial ^{n}f}{\partial x_{i}\,...\partial x_{j}}}(a_{1},\dots ,a_{n})={\frac {\partial ^{n}f}{\partial x_{j}\,...\partial x_{i}}}(a_{1},\dots ,a_{n}).}
Sea
f
:
Ω
→
R
{\displaystyle f:\Omega \to \mathbb {R} }
Ω
⊆
R
2
{\displaystyle \Omega \subseteq \mathbb {R} ^{2}}
Ω
{\displaystyle \Omega }
f
∈
C
2
(
Ω
)
{\displaystyle f\in {\mathcal {C}}^{2}(\Omega )}
∂
2
f
∂
x
∂
y
=
∂
2
f
∂
y
∂
x
{\displaystyle {\frac {\partial ^{2}f}{\partial x\;\partial y}}={\frac {\partial ^{2}f}{\partial y\;\partial x}}}
Sea
p
=
(
x
0
,
y
0
)
∈
Ω
{\displaystyle p=(x_{0},y_{0})\in \Omega \,}
Y sean
ε
{\displaystyle \varepsilon \,}
δ
>
0
{\displaystyle \delta >0\,}
(
x
0
−
ε
,
x
0
+
ε
)
×
(
y
0
−
δ
,
y
0
+
δ
)
⊂
Ω
{\displaystyle (x_{0}-\varepsilon ,x_{0}+\varepsilon )\times (y_{0}-\delta ,y_{0}+\delta )\subset \Omega \,}
Ω
{\displaystyle \Omega \,}
R
2
{\displaystyle \mathbb {R} ^{2}\,}
Se definen dos funciones
F
{\displaystyle F\,}
G
{\displaystyle G\,}
F
:
(
−
ε
,
ε
)
⊂
R
⟶
R
{\displaystyle F:(-\varepsilon ,\varepsilon )\subset \mathbb {R} \longrightarrow \mathbb {R} \,}
G
:
(
−
δ
,
δ
)
⊂
R
⟶
R
{\displaystyle G:(-\delta ,\delta )\subset \mathbb {R} \longrightarrow \mathbb {R} \,}
de modo que:
F
(
t
)
=
f
(
x
0
+
t
,
y
0
+
s
)
−
f
(
x
0
+
t
,
y
0
)
∀
t
∈
(
−
ε
,
ε
)
{\displaystyle F(t)=f(x_{0}+t,y_{0}+s)-f(x_{0}+t,y_{0})\qquad \forall t\in (-\varepsilon ,\varepsilon )\,}
G
(
s
)
=
f
(
x
0
+
t
,
y
0
+
s
)
−
f
(
x
0
,
y
0
+
s
)
∀
s
∈
(
−
δ
,
δ
)
{\displaystyle G(s)=f(x_{0}+t,y_{0}+s)-f(x_{0},y_{0}+s)\qquad \forall s\in (-\delta ,\delta )\,}
teorema de Lagrange :
F
(
t
)
−
F
(
0
)
=
(
t
−
0
)
F
′
(
ξ
1
)
=
t
[
∂
f
∂
x
(
x
0
+
ξ
1
,
y
0
+
s
)
−
∂
f
∂
x
(
x
0
+
ξ
1
,
y
0
)
]
=
{\displaystyle F(t)-F(0)=(t-0)F'(\xi _{1})=t\left[{\frac {\partial f}{\partial x}}(x_{0}+\xi _{1},y_{0}+s)-{\frac {\partial f}{\partial x}}(x_{0}+\xi _{1},y_{0})\right]=}
=
t
s
∂
2
f
∂
y
∂
x
(
x
0
+
ξ
1
,
y
0
+
σ
1
)
{\displaystyle =ts{\frac {{\partial }^{2}f}{\partial y\partial x}}(x_{0}+\xi _{1},y_{0}+\sigma _{1})\,}
y análogamente:
G
(
s
)
−
G
(
0
)
=
s
t
∂
2
f
∂
x
∂
y
(
x
0
+
ξ
2
,
y
0
+
σ
2
)
{\displaystyle G(s)-G(0)=st{\frac {{\partial }^{2}f}{\partial x\partial y}}(x_{0}+\xi _{2},y_{0}+\sigma _{2})\,}
con
ξ
i
∈
(
0
,
t
)
{\displaystyle \xi _{i}\in (0,t)\,}
σ
i
∈
(
0
,
s
)
{\displaystyle \sigma _{i}\in (0,s)\,}
t
,
s
>
0
{\displaystyle t,s>0\,}
Luego haciendo tender
t
{\displaystyle t\,}
s
{\displaystyle s\,}
0
{\displaystyle 0\,}
![{\displaystyle F(t)-F(0)=(t-0)F'(\xi _{1})=t\left[{\frac {\partial f}{\partial x}}(x_{0}+\xi _{1},y_{0}+s)-{\frac {\partial f}{\partial x}}(x_{0}+\xi _{1},y_{0})\right]=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49086a25922e5ac7e9fa7faa5950ddf8c65bc18f)
Datos: Q1503239