La derivata direzionale di una funzione scalare
f
(
x
)
=
f
(
x
1
,
x
2
,
…
,
x
n
)
{\displaystyle f(\mathbf {x} )=f(x_{1},x_{2},\ldots ,x_{n})}
vettore unitario
v
=
(
v
1
,
…
,
v
n
)
{\displaystyle \mathbf {v} =(v_{1},\ldots ,v_{n})}
limite :
D
v
f
(
x
)
=
lim
h
→
0
f
(
x
+
h
v
)
−
f
(
x
)
h
.
{\displaystyle D_{\mathbf {v} }{f}(\mathbf {x} )=\lim _{h\rightarrow 0}{\frac {f(\mathbf {x} +h\mathbf {v} )-f(\mathbf {x} )}{h}}.}
In ogni punto
x
{\displaystyle \mathbf {x} }
D
v
f
(
x
)
{\displaystyle D_{\mathbf {v} }{f}(\mathbf {x} )}
f
{\displaystyle f}
v
{\displaystyle \mathbf {v} }
Ad esempio, si consideri una funzione di due variabili
f
:
Ω
→
R
{\displaystyle f:\Omega \to \mathbb {R} }
Ω
⊆
R
2
{\displaystyle \Omega \subseteq \mathbb {R} ^{2}}
v
=
(
v
1
,
v
2
)
{\displaystyle \mathbf {v} =(v_{1},v_{2})}
f
{\displaystyle f\;}
v
{\displaystyle \mathbf {v} }
(
x
0
,
y
0
)
∈
Ω
{\displaystyle (x_{0},y_{0})\in \Omega }
D
v
f
(
x
0
,
y
0
)
=
lim
h
→
0
f
(
x
0
+
h
v
1
,
y
0
+
h
v
2
)
−
f
(
x
0
,
y
0
)
h
{\displaystyle D_{\mathbf {v} }f(x_{0},y_{0})=\lim _{h\to 0}{\frac {f(x_{0}+hv_{1},y_{0}+hv_{2})-f(x_{0},y_{0})}{h}}}
ed esiste se il limite è finito.
Se la funzione
f
{\displaystyle f}
differenziabile in
x
{\displaystyle \mathbf {x} }
v
{\displaystyle \mathbf {v} }
[1]
D
v
f
(
x
)
=
∇
f
(
x
)
⋅
v
,
{\displaystyle D_{\mathbf {v} }{f}(\mathbf {x} )=\nabla f(\mathbf {x} )\cdot \mathbf {v} ,}
dove
∇
{\displaystyle \nabla }
gradiente , e
⋅
{\displaystyle \cdot }
prodotto scalare euclideo .
Derivata direzionale di Dini
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Geometria differenziale
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Si può estendere il concetto di derivata direzionale presente nell'ordinario spazio euclideo ad una varietà differenziabile arbitraria tramite la derivata covariante, che consente di calcolare la derivata di un campo vettoriale , o di un più generale campo tensoriale , in un punto della varietà lungo una direzione fissata.
Sia
M
{\displaystyle M}
p
{\displaystyle \mathbf {p} }
M
{\displaystyle M}
f
{\displaystyle f}
p
{\displaystyle \mathbf {p} }
differenziabile in
p
{\displaystyle \mathbf {p} }
v
{\displaystyle \mathbf {v} }
vettore tangente
M
{\displaystyle M}
p
{\displaystyle \mathbf {p} }
γ
:
[
−
1
,
1
]
→
M
{\displaystyle \gamma :[-1,1]\to M}
γ
(
0
)
=
p
{\displaystyle \gamma (0)=\mathbf {p} }
γ
′
(
0
)
=
v
{\displaystyle \gamma '(0)=\mathbf {v} }
f
{\displaystyle f}
v
{\displaystyle \mathbf {v} }
∇
v
f
(
p
)
{\displaystyle \nabla _{\mathbf {v} }f(\mathbf {p} )}
∇
v
f
(
p
)
=
d
d
τ
f
∘
γ
(
τ
)
|
τ
=
0
.
{\displaystyle \nabla _{\mathbf {v} }f(\mathbf {p} )=\left.{\frac {d}{d\tau }}f\circ \gamma (\tau )\right|_{\tau =0}.}
Tale relazione è il punto di partenza anche per le definizioni di derivata di Lie e derivata esterna , centrali in geometria differenziale e topologia differenziale .
La nozione di derivata covariante è essenzialmente equivalente a quella di connessione : su una varietà differenziabile è possibile scegliere fra una infinità di possibili connessioni, e quindi di possibili nozioni di derivata covariante. Attraverso di essa, in fisica , si definiscono vari tensori che misurano la curvatura di una varietà, come il tensore di Riemann ed il tensore di Ricci .
Meccanica del continuo
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Molti importanti risultati della meccanica del continuo sono espressi tramite il concetto di derivata di vettori rispetto a vettori, e di tensori rispetto a vettori e tensori.[2]
Funzione scalare di vettori
modifica
Sia
f
(
v
)
{\displaystyle f(\mathbf {v} )}
v
{\displaystyle \mathbf {v} }
f
(
v
)
{\displaystyle f(\mathbf {v} )}
v
{\displaystyle \mathbf {v} }
v
{\displaystyle \mathbf {v} }
u
{\displaystyle \mathbf {u} }
∂
f
∂
v
⋅
u
=
D
f
(
v
)
[
u
]
=
[
d
d
α
f
(
v
+
α
u
)
]
α
=
0
,
∀
u
{\displaystyle {\frac {\partial f}{\partial \mathbf {v} }}\cdot \mathbf {u} =Df(\mathbf {v} )[\mathbf {u} ]=\left[{\frac {d}{d\alpha }}~f(\mathbf {v} +\alpha ~\mathbf {u} )\right]_{\alpha =0},\qquad \forall \mathbf {u} }
e gode delle seguenti proprietà:
Se
f
(
v
)
=
f
1
(
v
)
+
f
2
(
v
)
,
{\displaystyle f(\mathbf {v} )=f_{1}(\mathbf {v} )+f_{2}(\mathbf {v} ),}
∂
f
∂
v
⋅
u
=
(
∂
f
1
∂
v
+
∂
f
2
∂
v
)
⋅
u
.
{\displaystyle {\frac {\partial f}{\partial \mathbf {v} }}\cdot \mathbf {u} =\left({\frac {\partial f_{1}}{\partial \mathbf {v} }}+{\frac {\partial f_{2}}{\partial \mathbf {v} }}\right)\cdot \mathbf {u} .}
Se
f
(
v
)
=
f
1
(
v
)
f
2
(
v
)
,
{\displaystyle f(\mathbf {v} )=f_{1}(\mathbf {v} )~f_{2}(\mathbf {v} ),}
∂
f
∂
v
⋅
u
=
(
∂
f
1
∂
v
⋅
u
)
f
2
(
v
)
+
f
1
(
v
)
(
∂
f
2
∂
v
⋅
u
)
.
{\displaystyle {\frac {\partial f}{\partial \mathbf {v} }}\cdot \mathbf {u} =\left({\frac {\partial f_{1}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right)~f_{2}(\mathbf {v} )+f_{1}(\mathbf {v} )~\left({\frac {\partial f_{2}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right).}
Se
f
(
v
)
=
f
1
(
f
2
(
v
)
)
,
{\displaystyle f(\mathbf {v} )=f_{1}(f_{2}(\mathbf {v} )),}
∂
f
∂
v
⋅
u
=
∂
f
1
∂
f
2
∂
f
2
∂
v
⋅
u
.
{\displaystyle {\frac {\partial f}{\partial \mathbf {v} }}\cdot \mathbf {u} ={\frac {\partial f_{1}}{\partial f_{2}}}~{\frac {\partial f_{2}}{\partial \mathbf {v} }}\cdot \mathbf {u} .}
Funzione vettoriale di vettori
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Sia
f
(
v
)
{\displaystyle \mathbf {f} (\mathbf {v} )}
v
{\displaystyle \mathbf {v} }
f
(
v
)
{\displaystyle \mathbf {f} (\mathbf {v} )}
v
{\displaystyle \mathbf {v} }
v
{\displaystyle \mathbf {v} }
u
{\displaystyle \mathbf {u} }
∂
f
∂
v
⋅
u
=
D
f
(
v
)
[
u
]
=
[
d
d
α
f
(
v
+
α
u
)
]
α
=
0
,
∀
u
{\displaystyle {\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}\cdot \mathbf {u} =D\mathbf {f} (\mathbf {v} )[\mathbf {u} ]=\left[{\frac {d}{d\alpha }}~\mathbf {f} (\mathbf {v} +\alpha ~\mathbf {u} )\right]_{\alpha =0},\qquad \forall \mathbf {u} }
e gode delle seguenti proprietà:
Se
f
(
v
)
=
f
1
(
v
)
+
f
2
(
v
)
,
{\displaystyle \mathbf {f} (\mathbf {v} )=\mathbf {f} _{1}(\mathbf {v} )+\mathbf {f} _{2}(\mathbf {v} ),}
∂
f
∂
v
⋅
u
=
(
∂
f
1
∂
v
+
∂
f
2
∂
v
)
⋅
u
.
{\displaystyle {\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}\cdot \mathbf {u} =\left({\frac {\partial \mathbf {f} _{1}}{\partial \mathbf {v} }}+{\frac {\partial \mathbf {f} _{2}}{\partial \mathbf {v} }}\right)\cdot \mathbf {u} .}
Se
f
(
v
)
=
f
1
(
v
)
×
f
2
(
v
)
,
{\displaystyle \mathbf {f} (\mathbf {v} )=\mathbf {f} _{1}(\mathbf {v} )\times \mathbf {f} _{2}(\mathbf {v} ),}
∂
f
∂
v
⋅
u
=
(
∂
f
1
∂
v
⋅
u
)
×
f
2
(
v
)
+
f
1
(
v
)
×
(
∂
f
2
∂
v
⋅
u
)
.
{\displaystyle {\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}\cdot \mathbf {u} =\left({\frac {\partial \mathbf {f} _{1}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right)\times \mathbf {f} _{2}(\mathbf {v} )+\mathbf {f} _{1}(\mathbf {v} )\times \left({\frac {\partial \mathbf {f} _{2}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right).}
Se
f
(
v
)
=
f
1
(
f
2
(
v
)
)
,
{\displaystyle \mathbf {f} (\mathbf {v} )=\mathbf {f} _{1}(\mathbf {f} _{2}(\mathbf {v} )),}
∂
f
∂
v
⋅
u
=
∂
f
1
∂
f
2
⋅
(
∂
f
2
∂
v
⋅
u
)
.
{\displaystyle {\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}\cdot \mathbf {u} ={\frac {\partial \mathbf {f} _{1}}{\partial \mathbf {f} _{2}}}\cdot \left({\frac {\partial \mathbf {f} _{2}}{\partial \mathbf {v} }}\cdot \mathbf {u} \right).}
Funzione scalare di tensori di ordine 2
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Sia
f
(
S
)
{\displaystyle f({\boldsymbol {S}})}
S
{\displaystyle {\boldsymbol {S}}}
f
(
S
)
{\displaystyle f({\boldsymbol {S}})}
S
{\displaystyle {\boldsymbol {S}}}
S
{\displaystyle {\boldsymbol {S}}}
T
{\displaystyle {\boldsymbol {T}}}
∂
f
∂
S
:
T
=
D
f
(
S
)
[
T
]
=
[
d
d
α
f
(
S
+
α
T
)
]
α
=
0
,
{\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=Df({\boldsymbol {S}})[{\boldsymbol {T}}]=\left[{\frac {d}{d\alpha }}~f({\boldsymbol {S}}+\alpha {\boldsymbol {T}})\right]_{\alpha =0},}
per ogni tensore del secondo ordine
T
{\displaystyle {\boldsymbol {T}}}
Se
f
(
S
)
=
f
1
(
S
)
+
f
2
(
S
)
,
{\displaystyle f({\boldsymbol {S}})=f_{1}({\boldsymbol {S}})+f_{2}({\boldsymbol {S}}),}
∂
f
∂
S
:
T
=
(
∂
f
1
∂
S
+
∂
f
2
∂
S
)
:
T
.
{\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=\left({\frac {\partial f_{1}}{\partial {\boldsymbol {S}}}}+{\frac {\partial f_{2}}{\partial {\boldsymbol {S}}}}\right):{\boldsymbol {T}}.}
Se
f
(
S
)
=
f
1
(
S
)
f
2
(
S
)
,
{\displaystyle f({\boldsymbol {S}})=f_{1}({\boldsymbol {S}})~f_{2}({\boldsymbol {S}}),}
∂
f
∂
S
:
T
=
(
∂
f
1
∂
S
:
T
)
f
2
(
S
)
+
f
1
(
S
)
(
∂
f
2
∂
S
:
T
)
.
{\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=\left({\frac {\partial f_{1}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)~f_{2}({\boldsymbol {S}})+f_{1}({\boldsymbol {S}})~\left({\frac {\partial f_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right).}
Se
f
(
S
)
=
f
1
(
f
2
(
S
)
)
,
{\displaystyle f({\boldsymbol {S}})=f_{1}(f_{2}({\boldsymbol {S}})),}
∂
f
∂
S
:
T
=
∂
f
1
∂
f
2
(
∂
f
2
∂
S
:
T
)
.
{\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}={\frac {\partial f_{1}}{\partial f_{2}}}~\left({\frac {\partial f_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right).}
Funzione tensoriale di tensori di ordine 2
modifica
Sia
F
(
S
)
{\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})}
S
{\displaystyle {\boldsymbol {S}}}
F
(
S
)
{\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})}
S
{\displaystyle {\boldsymbol {S}}}
S
{\displaystyle {\boldsymbol {S}}}
T
{\displaystyle {\boldsymbol {T}}}
∂
F
∂
S
:
T
=
D
F
(
S
)
[
T
]
=
[
d
d
α
F
(
S
+
α
T
)
]
α
=
0
,
{\displaystyle {\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=D{\boldsymbol {F}}({\boldsymbol {S}})[{\boldsymbol {T}}]=\left[{\frac {d}{d\alpha }}~{\boldsymbol {F}}({\boldsymbol {S}}+\alpha {\boldsymbol {T}})\right]_{\alpha =0},}
per ogni tensore del secondo ordine
T
{\displaystyle {\boldsymbol {T}}}
Se
F
(
S
)
=
F
1
(
S
)
+
F
2
(
S
)
,
{\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})={\boldsymbol {F}}_{1}({\boldsymbol {S}})+{\boldsymbol {F}}_{2}({\boldsymbol {S}}),}
∂
F
∂
S
:
T
=
(
∂
F
1
∂
S
+
∂
F
2
∂
S
)
:
T
.
{\displaystyle {\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=\left({\frac {\partial {\boldsymbol {F}}_{1}}{\partial {\boldsymbol {S}}}}+{\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}\right):{\boldsymbol {T}}.}
Se
F
(
S
)
=
F
1
(
S
)
⋅
F
2
(
S
)
,
{\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})={\boldsymbol {F}}_{1}({\boldsymbol {S}})\cdot {\boldsymbol {F}}_{2}({\boldsymbol {S}}),}
∂
F
∂
S
:
T
=
(
∂
F
1
∂
S
:
T
)
⋅
F
2
(
S
)
+
F
1
(
S
)
⋅
(
∂
F
2
∂
S
:
T
)
.
{\displaystyle {\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=\left({\frac {\partial {\boldsymbol {F}}_{1}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)\cdot {\boldsymbol {F}}_{2}({\boldsymbol {S}})+{\boldsymbol {F}}_{1}({\boldsymbol {S}})\cdot \left({\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right).}
Se
F
(
S
)
=
F
1
(
F
2
(
S
)
)
,
{\displaystyle {\boldsymbol {F}}({\boldsymbol {S}})={\boldsymbol {F}}_{1}({\boldsymbol {F}}_{2}({\boldsymbol {S}})),}
∂
F
∂
S
:
T
=
∂
F
1
∂
F
2
:
(
∂
F
2
∂
S
:
T
)
.
{\displaystyle {\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}={\frac {\partial {\boldsymbol {F}}_{1}}{\partial {\boldsymbol {F}}_{2}}}:\left({\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right).}
Se
f
(
S
)
=
f
1
(
F
2
(
S
)
)
,
{\displaystyle f({\boldsymbol {S}})=f_{1}({\boldsymbol {F}}_{2}({\boldsymbol {S}})),}
∂
f
∂
S
:
T
=
∂
f
1
∂
F
2
:
(
∂
F
2
∂
S
:
T
)
.
{\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}={\frac {\partial f_{1}}{\partial {\boldsymbol {F}}_{2}}}:\left({\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right).}
^ W. Rudin , Pag. 219^ J. E. Marsden and T. J. R. Hughes, 2000, Mathematical Foundations of Elasticity , Dover.
Nicola Fusco , Paolo Marcellini , Carlo Sbordone , Lezioni di Analisi Matematica Due , Zanichelli, 2020, ISBN 9788808520203 , capitolo 3. Walter Rudin, Principi di analisi matematica , Milano, McGraw-Hill, 1991, ISBN 88-386-0647-1 . (EN Advanced Calculus for Applications ISBN 0-13-011189-9 . (EN Mathematical methods for physics and engineering , Cambridge University Press, 2010, ISBN 978-0-521-86153-3 . Collegamenti esterni
modifica
![{\displaystyle \gamma :[-1,1]\to M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9abbbb42b8959e778106bf7ae986e24f52d7d3d)
![{\displaystyle {\frac {\partial f}{\partial \mathbf {v} }}\cdot \mathbf {u} =Df(\mathbf {v} )[\mathbf {u} ]=\left[{\frac {d}{d\alpha }}~f(\mathbf {v} +\alpha ~\mathbf {u} )\right]_{\alpha =0},\qquad \forall \mathbf {u} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1ad3c9231f1fa4150a61bb5f4a5c395acf75ffa)
![{\displaystyle {\frac {\partial \mathbf {f} }{\partial \mathbf {v} }}\cdot \mathbf {u} =D\mathbf {f} (\mathbf {v} )[\mathbf {u} ]=\left[{\frac {d}{d\alpha }}~\mathbf {f} (\mathbf {v} +\alpha ~\mathbf {u} )\right]_{\alpha =0},\qquad \forall \mathbf {u} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/d3c741dc1d811f1e6aa4da1bf7e7fb4b5b439b3d)
![{\displaystyle {\frac {\partial f}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=Df({\boldsymbol {S}})[{\boldsymbol {T}}]=\left[{\frac {d}{d\alpha }}~f({\boldsymbol {S}}+\alpha {\boldsymbol {T}})\right]_{\alpha =0},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/daf66c2f82a00eb0d9e41e7f249411b806232a25)
![{\displaystyle {\frac {\partial {\boldsymbol {F}}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}=D{\boldsymbol {F}}({\boldsymbol {S}})[{\boldsymbol {T}}]=\left[{\frac {d}{d\alpha }}~{\boldsymbol {F}}({\boldsymbol {S}}+\alpha {\boldsymbol {T}})\right]_{\alpha =0},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b97b9743b0f0dd1ca35506661b6559a5c241705d)
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