O cálculo multivariábel (ou cálculo en varias variábeis) é máis a extensión do cálculo infinitesimal a funcións escalares e vectoriais de varias variábeis .[ 1]
Campo escalar con dúas variábeis Formularanse as definicións para campos vectoriais , que tamén son válidas para campos escalares. Sexa
f
:
V
⟶
W
{\displaystyle \mathbf {f} :V\longrightarrow W}
un campo vectorial que fai corresponder a todo punto P definido biunivocamente polo seu vector posición un vector
f
(
O
P
)
{\displaystyle \mathbf {f} {\big (}\mathbf {OP} {\big )}}
O é a orixe de coordenadas.
V
⊆
R
n
,
W
⊆
R
m
,
{\displaystyle V\subseteq \mathbb {R} ^{n},W\subseteq \mathbb {R} ^{m},}
n
>
1
{\displaystyle n>1}
m
⩾
1
{\displaystyle m\geqslant 1}
m
=
1
{\displaystyle m=1}
m
>
1
{\displaystyle m>1}
norma euclidiana para achar a magnitude dos vectores .Sexan
a
∈
R
n
{\displaystyle \mathbf {a} \in \mathbb {R} ^{n}}
b
∈
R
m
{\displaystyle \mathbf {b} \in \mathbb {R} ^{m}}
lim
x
→
a
f
(
x
)
=
b
{\displaystyle \lim _{\mathbf {x} \to \mathbf {a} }\mathbf {f} {\big (}\mathbf {x} {\big )}=\mathbf {b} }
ou ben,
f
(
x
)
→
b
{\displaystyle \mathbf {f} (\mathbf {x} )\rightarrow \mathbf {b} }
x
→
a
{\displaystyle \mathbf {x} \rightarrow \mathbf {a} }
para expresar o seguinte:
lim
‖
x
−
a
‖
→
0
‖
f
(
x
)
−
b
‖
=
0
{\displaystyle \lim _{{\big \|}\mathbf {x-a} {\big \|}\to 0}{\big \|}\mathbf {f} {\big (}\mathbf {x} {\big )}-\mathbf {b} {\big \|}=0}
onde
‖
x
‖
{\displaystyle {\big \|}\mathbf {x} {\big \|}}
norma euclidiana de
x
{\displaystyle \mathbf {x} }
x
=
(
x
1
,
…
,
x
n
)
,
a
=
(
a
1
,
…
,
a
n
)
,
{\displaystyle \mathbf {x} ={\big (}x_{1},\ldots ,x_{n}{\big )},\mathbf {a} ={\big (}a_{1},\ldots ,a_{n}{\big )},}
lim
(
x
1
,
…
,
x
n
)
→
(
a
1
,
…
,
a
n
)
f
(
x
1
,
…
,
x
n
)
=
b
{\displaystyle \lim _{{\big (}x_{1},\ldots ,x_{n}{\big )}\to {\big (}a_{1},\ldots ,a_{n}{\big )}}\mathbf {f} {\big (}x_{1},\ldots ,x_{n}{\big )}=\mathbf {b} }
ou, de forma equivalente,
lim
x
→
a
f
(
x
)
=
b
{\displaystyle \lim _{\mathbf {x} \to \mathbf {a} }\mathbf {f} {\big (}\mathbf {x} {\big )}=\mathbf {b} }
Dise que unha función
f
{\displaystyle \mathbf {f} }
é
continua en
a
⇔
lim
x
→
a
f
(
x
)
=
f
(
a
)
{\displaystyle \mathbf {a} \Leftrightarrow \lim _{\mathbf {x} \to \mathbf {a} }\mathbf {f} {\big (}\mathbf {x} {\big )}=\mathbf {f} {\big (}\mathbf {a} {\big )}}
Sexa
f
:
S
⊆
R
n
⟶
R
{\displaystyle f:S\subseteq \mathbb {R} ^{n}\longrightarrow \mathbb {R} }
x
{\displaystyle \mathbf {x} }
∈
S
,
{\displaystyle \in S,}
y
{\displaystyle \mathbf {y} }
R
n
{\displaystyle \mathbb {R} ^{n}}
derivada de f en
x
{\displaystyle \mathbf {x} }
y
{\displaystyle \mathbf {y} }
f
′
(
x
;
y
)
=
lim
h
→
0
f
(
x
+
h
y
)
−
f
(
x
)
h
{\displaystyle f'{\big (}\mathbf {x} ;\mathbf {y} {\big )}=\lim _{h\to 0}{\cfrac {f{\big (}\mathbf {x} +h\mathbf {y} {\big )}-f{\big (}\mathbf {x} {\big )}}{h}}}
∂
f
∂
x
k
=
lim
h
→
0
f
(
x
1
,
…
,
x
k
+
h
,
…
,
x
n
)
−
f
(
x
1
,
…
,
x
k
,
…
,
x
n
)
h
{\displaystyle {\cfrac {\partial f}{\partial x_{k}}}=\lim _{h\to 0}{\cfrac {f{\big (}x_{1},\ldots ,x_{k}+h,\ldots ,x_{n}{\big )}-f{\big (}x_{1},\ldots ,x_{k},\ldots ,x_{n}{\big )}}{h}}}
Se se deriva a expresión anterior respecto dunha segunda variábel,
x
j
{\displaystyle x_{j}}
∂
2
f
∂
x
j
∂
x
k
{\displaystyle {\cfrac {\partial ^{2}f}{\partial x_{j}\partial x_{k}}}}
∂
f
∂
x
k
{\displaystyle {\cfrac {\partial f}{\partial x_{k}}}}
x
k
{\displaystyle x_{k}}
x
j
,
∀
j
≠
k
{\displaystyle x_{j},\quad \forall j\neq k}
Dise que f é diferenciábel en
a
⇔
{\displaystyle \mathbf {a} \Leftrightarrow }
∃
f
L
:
R
n
⟶
R
|
lim
‖
v
‖
→
0
f
(
a
+
v
)
=
f
(
a
)
+
f
L
(
v
)
{\displaystyle \exists f_{L}:\mathbb {R} ^{n}\longrightarrow \mathbb {R} {\Big |}\lim _{{\big \|}\mathbf {v} {\big \|}\to \mathbf {0} }f{\big (}\mathbf {a} +\mathbf {v} {\big )}=f{\big (}\mathbf {a} {\big )}+f_{L}{\big (}\mathbf {v} {\big )}}
f
L
{\displaystyle f_{L}}
aplicación linear , que se define como a diferencial de f en a .A anterior ecuación é a
{\displaystyle }
f
(
a
+
v
)
{\displaystyle f{\big (}\mathbf {a} +\mathbf {v} {\big )}}
f
{\displaystyle f}
x
{\displaystyle \mathbf {x} }
f
L
(
y
)
⇒
{\displaystyle f_{L}{\big (}\mathbf {y} {\big )}\Rightarrow }
a)
∃
f
′
(
x
;
y
)
∀
y
∈
R
n
{\displaystyle \exists f'{\big (}\mathbf {x} ;\mathbf {y} {\big )}\quad \forall \mathbf {y} \in \mathbb {R} ^{n}}
b)
f
′
(
x
;
y
)
=
∑
k
=
1
n
y
k
∂
f
∂
x
k
{\displaystyle f'{\big (}\mathbf {x} ;\mathbf {y} {\big )}=\sum _{k=1}^{n}y_{k}{\cfrac {\partial f}{\partial x_{k}}}}
Sexa
f
:
S
⊂
R
n
⟶
R
{\displaystyle f:S\subset \mathbb {R} ^{n}\longrightarrow \mathbb {R} }
x
:
J
∈
R
⟶
S
{\displaystyle \mathbf {x} :J\in \mathbb {R} \longrightarrow S}
g
=
f
∘
x
{\displaystyle g=f\circ \mathbf {x} }
g
(
t
)
=
f
[
x
(
t
)
]
{\displaystyle g(t)=f{\Big [}\mathbf {x} {\big (}t{\big )}{\Big ]}}
g
′
(
t
)
=
∑
k
=
1
n
∂
f
∂
x
k
⋅
d
x
k
d
t
{\displaystyle \quad g'{\big (}t{\big )}=\sum _{k=1}^{n}{\cfrac {\partial f}{\partial x_{k}}}\cdot {\cfrac {dx_{k}}{dt}}}
Sexa
f
:
S
⊆
R
n
⟶
R
m
{\displaystyle \mathbf {f} :S\subseteq \mathbb {R} ^{n}\longrightarrow \mathbb {R} ^{m}}
x
∈
S
{\displaystyle \mathbf {x} \in S}
y
{\displaystyle \mathbf {y} }
f
′
(
x
;
y
)
=
lim
h
→
0
f
(
x
+
h
y
)
−
f
(
x
)
h
{\displaystyle \mathbf {f'} {\big (}\mathbf {x} ;\mathbf {y} {\big )}=\lim _{h\to 0}{\cfrac {\mathbf {f} {\big (}\mathbf {x} +h\mathbf {y} {\big )}-\mathbf {f} {\big (}\mathbf {x} {\big )}}{h}}}
Expresando
f
′
(
x
;
y
)
{\displaystyle \mathbf {f'} {\big (}\mathbf {x} ;\mathbf {y} {\big )}}
f
′
(
x
;
y
)
=
[
f
1
′
(
x
;
y
)
,
…
,
f
m
′
(
x
;
y
)
]
{\displaystyle \mathbf {f'} {\big (}\mathbf {x} ;\mathbf {y} {\big )}={\Big [}f'_{1}{\big (}\mathbf {x} ;\mathbf {y} {\big )},\ldots ,f'_{m}{\big (}\mathbf {x} ;\mathbf {y} {\big )}{\Big ]}}
Dise que
f
{\displaystyle \mathbf {f} }
⇔
∃
f
L
:
R
n
⟶
R
m
{\displaystyle \Leftrightarrow \exists \mathbf {f} _{L}:\mathbb {R} ^{n}\longrightarrow \mathbb {R} ^{m}}
aplicación linear que verifica:
lim
‖
v
‖
→
0
f
(
x
+
v
)
=
f
(
x
)
+
f
L
(
v
)
{\displaystyle \lim _{{\big \|}\mathbf {v} {\big \|}\to 0}\mathbf {f} {\big (}\mathbf {x} +\mathbf {v} {\big )}=\mathbf {f} {\big (}\mathbf {x} {\big )}+\mathbf {f} _{L}{\big (}\mathbf {v} {\big )}}
Esta é a fórmula de Taylor de primeira orde para
f
.
f
L
(
v
)
=
f
′
(
x
;
v
)
{\displaystyle \mathbf {f} .\quad \mathbf {f} _{L}{\big (}\mathbf {v} {\big )}=\mathbf {f} '{\big (}\mathbf {x} ;\mathbf {v} {\big )}}
A matriz de
f
′
{\displaystyle \mathbf {f} '}
matriz jacobiana .
Se un campo vectorial
f
{\displaystyle \mathbf {f} }
diferenciábel en
x
⇒
{\displaystyle \mathbf {x} \Rightarrow }
continuo en
x
{\displaystyle \mathbf {x} }
Sexa
h
(
x
)
=
(
f
∘
g
)
(
x
)
{\displaystyle \mathbf {h} {\big (}\mathbf {x} {\big )}={\big (}\mathbf {f} \circ \mathbf {g} {\big )}{\big (}\mathbf {x} {\big )}}
x
{\displaystyle \mathbf {x} }
h
′
(
x
)
{\displaystyle \mathbf {h} '{\big (}\mathbf {x} {\big )}}
h
′
(
x
)
=
f
′
[
g
(
x
)
]
∘
g
′
(
x
)
{\displaystyle \mathbf {h} '{\big (}\mathbf {x} {\big )}=\mathbf {f} '{\Big [}\mathbf {g} {\big (}\mathbf {x} {\big )}{\Big ]}\circ \mathbf {g} '{\big (}\mathbf {x} {\big )}}
∂
2
f
∂
x
i
∂
x
j
=
∂
2
f
∂
x
j
∂
x
i
∀
i
≠
j
⇔
{\displaystyle {\cfrac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}={\cfrac {\partial ^{2}f}{\partial x_{j}\partial x_{i}}}\quad \forall i\neq j\Leftrightarrow }
derivadas parciais existen e son continuas en
x
{\displaystyle \mathbf {x} }
Función cun punto de sela Defínense os seguintes conceptos:
Un campo escalar ten un máximo en
x
=
a
⇔
{\displaystyle \mathbf {x} =\mathbf {a} \Leftrightarrow }
n-bola
B
(
a
)
|
∀
x
∈
B
(
a
)
f
(
x
)
⩽
f
(
a
)
{\displaystyle B{\big (}\mathbf {a} {\big )}{\Big |}\forall \mathbf {x} \in B{\big (}\mathbf {a} {\big )}\quad f{\big (}\mathbf {x} {\big )}\leqslant f{\big (}\mathbf {a} {\big )}}
Un campo escalar ten un mínimo en
x
=
a
⇔
{\displaystyle \mathbf {x} =\mathbf {a} \Leftrightarrow }
B
(
a
)
|
∀
x
∈
B
(
a
)
f
(
x
)
⩾
f
(
a
)
{\displaystyle B{\big (}\mathbf {a} {\big )}{\Big |}\forall \mathbf {x} \in B{\big (}\mathbf {a} {\big )}\quad f{\big (}\mathbf {x} {\big )}\geqslant f{\big (}\mathbf {a} {\big )}}
Un campo escalar ten un punto de sela
⇔
{\displaystyle \Leftrightarrow }
∀
B
(
a
)
∃
x
|
f
(
x
)
⩽
f
(
a
)
∧
∃
x
|
f
(
x
)
⩾
f
(
a
)
{\displaystyle \forall B{\big (}\mathbf {a} {\big )}\quad \exists \mathbf {x} {\big |}f{\big (}\mathbf {x} {\big )}\leqslant f{\big (}\mathbf {a} {\big )}\land \exists \mathbf {x} {\big |}f{\big (}\mathbf {x} {\big )}\geqslant f{\big (}\mathbf {a} {\big )}}
Para saber se é un dos casos anteriores:
Obtense
x
|
∂
f
∂
x
k
=
0
∀
k
|
1
⩽
k
⩽
n
{\displaystyle \mathbf {x} {\Big |}{\cfrac {\partial f}{\partial x_{k}}}=0\qquad \forall k{\Big |}1\leqslant k\leqslant n}
Obtense a matriz hessiana de f . Sexa esta
F
(
x
)
{\displaystyle \mathbf {F} {\big (}\mathbf {x} {\big )}}
F
(
x
)
{\displaystyle \mathbf {F} {\big (}\mathbf {x} {\big )}}
definida positiva
⇒
f
{\displaystyle \Rightarrow f}
mínimo relativo en
x
{\displaystyle \mathbf {x} }
F
(
x
)
{\displaystyle \mathbf {F} {\big (}\mathbf {x} {\big )}}
definida negativa
⇒
f
{\displaystyle \Rightarrow f}
máximo relativo en
x
{\displaystyle \mathbf {x} }
F
(
x
)
{\displaystyle \mathbf {F} {\big (}\mathbf {x} {\big )}}
indefinida
⇒
f
{\displaystyle \Rightarrow f}
punto de sela en
x
{\displaystyle \mathbf {x} }
No anterior supúxose que
∂
2
f
∂
x
i
∂
x
j
{\displaystyle {\cfrac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}}
∀
i
,
j
|
1
⩽
i
⩽
n
,
1
⩽
j
⩽
n
{\displaystyle \forall i,j{\big |}1\leqslant i\leqslant n,1\leqslant j\leqslant n}
↑ Richard Courant; Fritz John (14 December 1999). Introduction to Calculus and Analysis Volume II/2 . Springer Science & Business Media. ISBN 978-3-540-66570-0 .
![{\displaystyle \lim _{\mathbf {x} \to \mathbf {a} }{\big [}\mathbf {f} +\mathbf {g} {\big ]}{\big (}\mathbf {x} {\big )}=\mathbf {b} +\mathbf {c} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/97e7d295bd1d5497f0c98c4d16c52d2720d749d0)
![{\displaystyle {\big (}\mathbf {f} \circ \mathbf {g} {\big )}{\big (}\mathbf {x} {\big )}=\mathbf {f} {\Big [}\mathbf {g} {\big (}\mathbf {x} {\big )}{\Big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e050e088fa87cd14516180f245242f9c90b4d73)
![{\displaystyle g(t)=f{\Big [}\mathbf {x} {\big (}t{\big )}{\Big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c24d0a6ebe6f93545e22280032f94396d0320cf9)
![{\displaystyle \mathbf {f'} {\big (}\mathbf {x} ;\mathbf {y} {\big )}={\Big [}f'_{1}{\big (}\mathbf {x} ;\mathbf {y} {\big )},\ldots ,f'_{m}{\big (}\mathbf {x} ;\mathbf {y} {\big )}{\Big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b886201a23ee2c88781d76fe6abf5585db77bc24)
![{\displaystyle \mathbf {h} '{\big (}\mathbf {x} {\big )}=\mathbf {f} '{\Big [}\mathbf {g} {\big (}\mathbf {x} {\big )}{\Big ]}\circ \mathbf {g} '{\big (}\mathbf {x} {\big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ee4c8d71f25c0b0392d15f41ec44fa1b485ee56)
