El cálculu multivariable (o cálculu en delles variables) nun ye más que la estensión del cálculu infinitesimal a funciones angulares y vectoriales de delles variables.
Vamos Formular les definiciones pa campos vectoriales. Tamién van ser válides para campos angulares. Sía : f : V ⟶ W {\displaystyle \mathbf {f} :V\longrightarrow W} un campu vectorial que fai corresponder a tou puntu P definíu biunívocamente pol so vector posición un vector f ( O P ) {\displaystyle \mathbf {f} {\big (}\mathbf {OP} {\big )}} onde'l puntu O ye'l nuesu orixe de coordenaes.
Sean a ∈ R n {\displaystyle \mathbf {a} \in \mathbb {R} ^{n}} y b ∈ R m . {\displaystyle \mathbf {b} \in \mathbb {R} ^{m}.} Escribimos:
onde ‖ x ‖ {\displaystyle {\big \|}\mathbf {x} {\big \|}} ye la norma euclídea de x {\displaystyle \mathbf {x} } . Espresándolo en función de les componentes de 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 )},}
o, de forma equivalente, : lim x → a f ( x ) = b {\displaystyle \lim _{\mathbf {x} \to \mathbf {a} }\mathbf {f} {\big (}\mathbf {x} {\big )}=\mathbf {b} }
Dicimos qu'una función f {\displaystyle \mathbf {f} } ye 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 )}}
lim x → a f ( x ) = b , lim x → a g ( x ) = c ⇒ {\displaystyle \lim _{\mathbf {x} \to \mathbf {a} }\mathbf {f} {\big (}\mathbf {x} {\big )}=\mathbf {b} ,\lim _{\mathbf {x} \to \mathbf {a} }\mathbf {g} {\big (}\mathbf {x} {\big )}=\mathbf {c} \Rightarrow } a) lim x → a [ f + g ] ( x ) = b + c {\displaystyle \lim _{\mathbf {x} \to \mathbf {a} }{\big [}\mathbf {f} +\mathbf {g} {\big ]}{\big (}\mathbf {x} {\big )}=\mathbf {b} +\mathbf {c} } b) lim x → a λ f ( x ) = λ b ∀ λ ∈ R {\displaystyle \lim _{\mathbf {x} \to \mathbf {a} }\lambda \mathbf {f} {\big (}\mathbf {x} {\big )}=\lambda \mathbf {b} \quad \forall \lambda \in \mathbb {R} } c) lim x → a ( f ⋅ g ) ( x ) = b ⋅ c {\displaystyle \lim _{\mathbf {x} \to \mathbf {a} }{\big (}\mathbf {f} \cdot \mathbf {g} {\big )}{\big (}\mathbf {x} {\big )}=\mathbf {b} \cdot \mathbf {c} } (productu angular de b {\displaystyle \mathbf {b} } con c {\displaystyle \mathbf {c} } ). d) lim x → a ‖ f ( x ) ‖ = ‖ b ‖ {\displaystyle \lim _{\mathbf {x} \to \mathbf {a} }{\Big \|}\mathbf {f} {\big (}\mathbf {x} {\big )}{\Big \|}={\big \|}\mathbf {b} {\big \|}}
lim x → a f ( x ) = b , lim x → a g ( x ) = c ⇒ {\displaystyle \lim _{\mathbf {x} \to \mathbf {a} }\mathbf {f} {\big (}\mathbf {x} {\big )}=\mathbf {b} ,\lim _{\mathbf {x} \to \mathbf {a} }\mathbf {g} {\big (}\mathbf {x} {\big )}=\mathbf {c} \Rightarrow }
Sean f {\displaystyle \mathbf {f} } y g {\displaystyle \mathbf {g} } dos funciones tales que la función compuesta f ∘ g {\displaystyle \mathbf {f} \circ \mathbf {g} } ta definida en a {\displaystyle \mathbf {a} } , siendo : ( f ∘ g ) ( x ) = f [ g ( x ) ] {\displaystyle {\big (}\mathbf {f} \circ \mathbf {g} {\big )}{\big (}\mathbf {x} {\big )}=\mathbf {f} {\Big [}\mathbf {g} {\big (}\mathbf {x} {\big )}{\Big ]}} g {\displaystyle \mathbf {g} } ye continua en a {\displaystyle \mathbf {a} } y f {\displaystyle \mathbf {f} } ye continua en g ( a ) ⇒ ( f ∘ g ) {\displaystyle \mathbf {g} {\big (}\mathbf {a} {\big )}\Rightarrow {\big (}\mathbf {f} \circ \mathbf {g} {\big )}} ye continua en a {\displaystyle \mathbf {a} } .
Sean f {\displaystyle \mathbf {f} } y g {\displaystyle \mathbf {g} } dos funciones tales que la función compuesta f ∘ g {\displaystyle \mathbf {f} \circ \mathbf {g} } ta definida en a {\displaystyle \mathbf {a} } , siendo : ( f ∘ g ) ( x ) = f [ g ( x ) ] {\displaystyle {\big (}\mathbf {f} \circ \mathbf {g} {\big )}{\big (}\mathbf {x} {\big )}=\mathbf {f} {\Big [}\mathbf {g} {\big (}\mathbf {x} {\big )}{\Big ]}}
Sía f : S ⊆ R n ⟶ R {\displaystyle f:S\subseteq \mathbb {R} ^{n}\longrightarrow \mathbb {R} } . Sía x {\displaystyle \mathbf {x} } un vector que'l so orixe ye l'orixe de coordenaes y que'l so estremu ∈ S , {\displaystyle \in S,} y y {\displaystyle \mathbf {y} } un vector arbitrariu de R n {\displaystyle \mathbb {R} ^{n}} . Definimos la derivada de f en x {\displaystyle \mathbf {x} } al respective de y {\displaystyle \mathbf {y} } como : 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}}}
Dicimos que f ye diferenciable 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}} hai de ser una aplicación llinial, que definimos como la diferencial de f en a.
Dicimos que f ye diferenciable en a ⇔ {\displaystyle \mathbf {a} \Leftrightarrow }
f {\displaystyle f} ye diferenciable en x {\displaystyle \mathbf {x} } con diferencial 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}}}}
f {\displaystyle f} ye diferenciable en x {\displaystyle \mathbf {x} } con diferencial f L ( y ) ⇒ {\displaystyle f_{L}{\big (}\mathbf {y} {\big )}\Rightarrow }
Sía f : S ⊂ R n ⟶ R {\displaystyle f:S\subset \mathbb {R} ^{n}\longrightarrow \mathbb {R} } un campu angular y x : J ∈ R ⟶ S {\displaystyle \mathbf {x} :J\in \mathbb {R} \longrightarrow S} . Definimos la función compuesta g = f ∘ x {\displaystyle g=f\circ \mathbf {x} } como g ( t ) = f [ x ( t ) ] {\displaystyle g(t)=f{\Big [}\mathbf {x} {\big (}t{\big )}{\Big ]}} , entós 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}}}
Sía f : S ⊆ R n ⟶ R m {\displaystyle \mathbf {f} :S\subseteq \mathbb {R} ^{n}\longrightarrow \mathbb {R} ^{m}} un campu vectorial. Sía x ∈ S {\displaystyle \mathbf {x} \in S} y y {\displaystyle \mathbf {y} } un vector cualesquier. Definimos la derivada : 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}}}
Espresando f ′ ( x ; y ) {\displaystyle \mathbf {f'} {\big (}\mathbf {x} ;\mathbf {y} {\big )}} en función de los sos componentes, tenemos 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 ]}}
Dicimos que f {\displaystyle \mathbf {f} } ye diferenciable ⇔ ∃ f L : R n ⟶ R m {\displaystyle \Leftrightarrow \exists \mathbf {f} _{L}:\mathbb {R} ^{n}\longrightarrow \mathbb {R} ^{m}} , aplicación llinial 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 )}} .
Dicimos que f {\displaystyle \mathbf {f} } ye diferenciable ⇔ ∃ f L : R n ⟶ R m {\displaystyle \Leftrightarrow \exists \mathbf {f} _{L}:\mathbb {R} ^{n}\longrightarrow \mathbb {R} ^{m}} , aplicación llinial que verifica:
La matriz de f ′ {\displaystyle \mathbf {f} '} ye'l so matriz jacobiana.
Si un campu vectorial f {\displaystyle \mathbf {f} } ye diferenciable en x ⇒ {\displaystyle \mathbf {x} \Rightarrow } ye continuu en x {\displaystyle \mathbf {x} } .
Sía h ( x ) = ( f ∘ g ) ( x ) {\displaystyle \mathbf {h} {\big (}\mathbf {x} {\big )}={\big (}\mathbf {f} \circ \mathbf {g} {\big )}{\big (}\mathbf {x} {\big )}} un campu vectorial definíu y diferenciable en x {\displaystyle \mathbf {x} } . El so diferencial h ′ ( x ) {\displaystyle \mathbf {h} '{\big (}\mathbf {x} {\big )}} resulta ser 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 )}}
Sía h ( x ) = ( f ∘ g ) ( x ) {\displaystyle \mathbf {h} {\big (}\mathbf {x} {\big )}={\big (}\mathbf {f} \circ \mathbf {g} {\big )}{\big (}\mathbf {x} {\big )}} un campu vectorial definíu y diferenciable en x {\displaystyle \mathbf {x} } . El so diferencial h ′ ( x ) {\displaystyle \mathbf {h} '{\big (}\mathbf {x} {\big )}} resulta ser
∂ 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 } dambes derivaes parciales esisten y son continues en x {\displaystyle \mathbf {x} } .
Un campu angular tien un máximu en x = a ⇔ {\displaystyle \mathbf {x} =\mathbf {a} \Leftrightarrow } esiste una 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 campu angular tien un mínimu en x = a ⇔ {\displaystyle \mathbf {x} =\mathbf {a} \Leftrightarrow } esiste una 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 )}\geqslant f{\big (}\mathbf {a} {\big )}}
Un campu angular tien un puntu de ensilladura ⇔ {\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 )}} .
Un campu angular tien un puntu de ensilladura ⇔ {\displaystyle \Leftrightarrow }
Pa saber si ye unu de los casos anteriores:
No enantes espuesto, supunximos que ∂ 2 f ∂ x i ∂ x j {\displaystyle {\cfrac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}} ye continua ∀ i , j | 1 ⩽ i ⩽ n , 1 ⩽ j ⩽ n {\displaystyle \forall i,j{\big |}1\leqslant i\leqslant n,1\leqslant j\leqslant n}
y 
son funciones angulares. Por tanto, si :
tenemos
![{\displaystyle {\begin{array}{rl}a)&\mathbf {f} {\big (}\mathbf {x} )={\big [}f_{1}{\big (}\mathbf {x} {\big )},\ldots ,f_{m}{\big (}\mathbf {x} {\big )}{\big ]},\mathbf {g} {\big (}\mathbf {x} )={\Big [}g_{1}{\big (}\mathbf {x} {\big )},\ldots ,g_{m}{\big (}\mathbf {x} {\big )}{\Big ]}\\&\lim _{\mathbf {x} \to \mathbf {a} }{\big (}\mathbf {f} +\mathbf {g} {\big )}{\big (}\mathbf {x} {\big )}=\lim _{\mathbf {x} \to \mathbf {a} }{\Big [}{\big (}f_{1}+g_{1}{\big )}{\big (}\mathbf {x} {\big )},\ldots ,{\big (}f_{m}+g_{m}{\big )}{\big (}\mathbf {x} {\big )}{\Big ]}=\\&{\Big [}\lim _{\mathbf {x} \to \mathbf {a} }{\big (}f_{1}+g_{1}{\big )}{\big (}\mathbf {x} {\big )},\ldots ,\lim _{\mathbf {x} \to \mathbf {a} }{\big (}f_{m}+g_{m}{\big )}{\big (}\mathbf {x} {\big )}{\Big ]}=\\&{\Big [}\lim _{\mathbf {x} \to \mathbf {a} }f_{1}{\big (}\mathbf {x} {\big )}+\lim _{\mathbf {x} \to \mathbf {a} }g_{1}(\mathbf {x} {\big )},\ldots ,\lim _{\mathbf {x} \to \mathbf {a} }f_{m}{\big (}\mathbf {x} {\big )}+\lim _{\mathbf {x} \to \mathbf {a} }g_{m}{\big (}\mathbf {x} {\big )}{\Big ]}=\\&{\big (}b_{1}+c_{1},\ldots ,b_{m}+c_{m}{\big )}={\big (}b_{1},\ldots ,b_{m}{\big )}+{\big (}c_{1},\ldots ,c_{m}{\big )}=\mathbf {b} +\mathbf {c} \end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95faf263a1baf0db2c17757d92d4652b8000a3c0)
![{\displaystyle {\begin{array}{rl}b)&\lim _{\mathbf {x} \to \mathbf {a} }\lambda \mathbf {f} {\big (}\mathbf {x} {\big )}=\lim _{\mathbf {x} \to \mathbf {a} }\lambda {\Big [}f_{1}{\big (}\mathbf {x} {\big )},\ldots ,f_{m}{\big (}\mathbf {x} {\big )}{\Big ]}=\lim _{\mathbf {x} \to \mathbf {a} }{\Big [}\lambda f_{1}{\big (}\mathbf {x} {\big )},\ldots ,\lambda f_{m}{\big (}\mathbf {x} {\big )}{\Big ]}=\\&{\Big [}\lim _{\mathbf {x} \to \mathbf {a} }\lambda f_{1}{\big (}\mathbf {x} {\big )},\ldots ,\lim _{\mathbf {x} \to \mathbf {a} }\lambda f_{m}{\big (}\mathbf {x} {\big )}{\Big ]}={\Big [}\lambda \lim _{\mathbf {x} \to \mathbf {a} }f_{1}{\big (}\mathbf {x} {\big )},\ldots ,\lambda \lim _{\mathbf {x} \to \mathbf {a} }f_{m}{\big (}\mathbf {x} {\big )}{\Big ]}=\\&\lambda {\Big [}\lim _{\mathbf {x} \to \mathbf {a} }f_{1}{\big (}\mathbf {x} {\big )},\ldots ,\lim _{\mathbf {x} \to \mathbf {a} }f_{m}{\big (}\mathbf {x} {\big )}{\Big ]}=\lambda {\big (}b_{1},\ldots ,b_{m}{\big )}=\lambda \mathbf {b} \end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10fef6b37ec01f2a38f73284854a3e4d6b80462c)
![{\displaystyle c)\quad {\big (}\mathbf {f} \cdot \mathbf {g} {\big )}{\big (}\mathbf {x} {\big )}-\mathbf {b} \cdot \mathbf {c} ={\Big [}\mathbf {f} {\big (}\mathbf {x} {\big )}-\mathbf {b} {\Big ]}\cdot {\Big [}\mathbf {g} {\big (}\mathbf {x} {\big )}-\mathbf {c} {\Big ]}+\mathbf {b} \cdot {\Big [}\mathbf {g} {\big (}\mathbf {x} {\big )}-\mathbf {c} {\Big ]}+\mathbf {c} \cdot {\Big [}\mathbf {f} {\big (}\mathbf {x} {\big )}-\mathbf {b} {\Big ]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/66af9fd2140f9780543400554816ae1043fbe68f)
, como queríamos demostrar.
![{\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 {\begin{array}{l}\lim _{{\big \|}\mathbf {x} -\mathbf {a} {\big \|}\to 0}{\Big \|}\mathbf {f} {\Big [}\mathbf {g} {\big (}\mathbf {x} {\big )}{\Big ]}-\mathbf {f} {\Big [}\mathbf {g} {\big (}\mathbf {a} {\big )}{\Big ]}{\Big \|}=\lim _{{\big \|}\mathbf {y} -\mathbf {b} {\big \|}\to 0}{\Big \|}\mathbf {f} {\big (}\mathbf {y} {\big )}-\mathbf {f} {\big (}\mathbf {b} {\big )}{\Big \|}=0\Rightarrow \\\lim _{\mathbf {x} \to \mathbf {a} }\mathbf {f} {\Big [}\mathbf {g} {\big (}\mathbf {x} {\big )}{\Big ]}=\mathbf {f} {\Big [}\mathbf {g} {\big (}\mathbf {a} {\big )}{\Big ]}\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e0972f84a181ff7e0e1de0688a2b517a3becc0b)
![{\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)
