Derivadas con respecto a vectores y tensores de segundo orden
editar
Gradiente de un campo tensorial
editar
El gradiente ,
∇
T
{\displaystyle {\boldsymbol {\nabla }}{\boldsymbol {T}}}
T
(
x
)
{\displaystyle {\boldsymbol {T}}(\mathbf {x} )}
c se define como:
∇
T
⋅
c
=
lim
α
→
0
d
d
α
T
(
x
+
α
c
)
{\displaystyle {\boldsymbol {\nabla }}{\boldsymbol {T}}\cdot \mathbf {c} =\lim _{\alpha \rightarrow 0}\quad {\cfrac {d}{d\alpha }}~{\boldsymbol {T}}(\mathbf {x} +\alpha \mathbf {c} )}
El gradiente de un campo tensorial de orden n es un campo tensorial de orden n +1.
Coordenadas cartesianas
editar
Si
e
1
,
e
2
,
e
3
{\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}}
coordenadas cartesianas , con las coordenadas de los puntos indicadas por (
x
1
,
x
2
,
x
3
{\displaystyle x_{1},x_{2},x_{3}}
T
{\displaystyle {\boldsymbol {T}}}
∇
T
=
∂
T
∂
x
i
⊗
e
i
{\displaystyle {\boldsymbol {\nabla }}{\boldsymbol {T}}={\cfrac {\partial {\boldsymbol {T}}}{\partial x_{i}}}\otimes \mathbf {e} _{i}}
Demostración
Los vectores x y c se pueden escribir como
x
=
x
i
e
i
{\displaystyle \mathbf {x} =x_{i}~\mathbf {e} _{i}}
c
=
c
i
e
i
{\displaystyle \mathbf {c} =c_{i}~\mathbf {e} _{i}}
y := x + αc . En ese caso, el gradiente viene dado por
∇
T
⋅
c
=
d
d
α
T
(
x
1
+
α
c
1
,
x
2
+
α
c
2
,
x
3
+
α
c
3
)
|
α
=
0
≡
d
d
α
T
(
y
1
,
y
2
,
y
3
)
|
α
=
0
=
[
∂
T
∂
y
1
∂
y
1
∂
α
+
∂
T
∂
y
2
∂
y
2
∂
α
+
∂
T
∂
y
3
∂
y
3
∂
α
]
α
=
0
=
[
∂
T
∂
y
1
c
1
+
∂
T
∂
y
2
c
2
+
∂
T
∂
y
3
c
3
]
α
=
0
=
∂
T
∂
x
1
c
1
+
∂
T
∂
x
2
c
2
+
∂
T
∂
x
3
c
3
≡
∂
T
∂
x
i
c
i
=
∂
T
∂
x
i
(
e
i
⋅
c
)
=
[
∂
T
∂
x
i
⊗
e
i
]
⋅
c
◻
{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}{\boldsymbol {T}}\cdot \mathbf {c} &=\left.{\cfrac {d}{d\alpha }}~{\boldsymbol {T}}(x_{1}+\alpha c_{1},x_{2}+\alpha c_{2},x_{3}+\alpha c_{3})\right|_{\alpha =0}\equiv \left.{\cfrac {d}{d\alpha }}~{\boldsymbol {T}}(y_{1},y_{2},y_{3})\right|_{\alpha =0}\\&=\left[{\cfrac {\partial {\boldsymbol {T}}}{\partial y_{1}}}~{\cfrac {\partial y_{1}}{\partial \alpha }}+{\cfrac {\partial {\boldsymbol {T}}}{\partial y_{2}}}~{\cfrac {\partial y_{2}}{\partial \alpha }}+{\cfrac {\partial {\boldsymbol {T}}}{\partial y_{3}}}~{\cfrac {\partial y_{3}}{\partial \alpha }}\right]_{\alpha =0}=\left[{\cfrac {\partial {\boldsymbol {T}}}{\partial y_{1}}}~c_{1}+{\cfrac {\partial {\boldsymbol {T}}}{\partial y_{2}}}~c_{2}+{\cfrac {\partial {\boldsymbol {T}}}{\partial y_{3}}}~c_{3}\right]_{\alpha =0}\\&={\cfrac {\partial {\boldsymbol {T}}}{\partial x_{1}}}~c_{1}+{\cfrac {\partial {\boldsymbol {T}}}{\partial x_{2}}}~c_{2}+{\cfrac {\partial {\boldsymbol {T}}}{\partial x_{3}}}~c_{3}\equiv {\cfrac {\partial {\boldsymbol {T}}}{\partial x_{i}}}~c_{i}={\cfrac {\partial {\boldsymbol {T}}}{\partial x_{i}}}~(\mathbf {e} _{i}\cdot \mathbf {c} )=\left[{\cfrac {\partial {\boldsymbol {T}}}{\partial x_{i}}}\otimes \mathbf {e} _{i}\right]\cdot \mathbf {c} \qquad \square \end{aligned}}}
Dado que los vectores de la base no varían en un sistema de coordenadas cartesiano, tenemos las siguientes relaciones para los gradientes de un campo escalar
ϕ
{\displaystyle \phi }
v' y un campo tensorial de segundo orden
S
{\displaystyle {\boldsymbol {S}}}
∇
ϕ
=
∂
ϕ
∂
x
i
e
i
=
ϕ
,
i
e
i
∇
v
=
∂
(
v
j
e
j
)
∂
x
i
⊗
e
i
=
∂
v
j
∂
x
i
e
j
⊗
e
i
=
v
j
,
i
e
j
⊗
e
i
∇
S
=
∂
(
S
j
k
e
j
⊗
e
k
)
∂
x
i
⊗
e
i
=
∂
S
j
k
∂
x
i
e
j
⊗
e
k
⊗
e
i
=
S
j
k
,
i
e
j
⊗
e
k
⊗
e
i
{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\phi &={\cfrac {\partial \phi }{\partial x_{i}}}~\mathbf {e} _{i}=\phi _{,i}~\mathbf {e} _{i}\\{\boldsymbol {\nabla }}\mathbf {v} &={\cfrac {\partial (v_{j}\mathbf {e} _{j})}{\partial x_{i}}}\otimes \mathbf {e} _{i}={\cfrac {\partial v_{j}}{\partial x_{i}}}~\mathbf {e} _{j}\otimes \mathbf {e} _{i}=v_{j,i}~\mathbf {e} _{j}\otimes \mathbf {e} _{i}\\{\boldsymbol {\nabla }}{\boldsymbol {S}}&={\cfrac {\partial (S_{jk}\mathbf {e} _{j}\otimes \mathbf {e} _{k})}{\partial x_{i}}}\otimes \mathbf {e} _{i}={\cfrac {\partial S_{jk}}{\partial x_{i}}}~\mathbf {e} _{j}\otimes \mathbf {e} _{k}\otimes \mathbf {e} _{i}=S_{jk,i}~\mathbf {e} _{j}\otimes \mathbf {e} _{k}\otimes \mathbf {e} _{i}\end{aligned}}}
editar
Si
g
1
,
g
2
,
g
3
{\displaystyle \mathbf {g} ^{1},\mathbf {g} ^{2},\mathbf {g} ^{3}}
base contravariante en un sistema de coordenadas curvilíneas , con las coordenadas de los puntos indicadas por (
ξ
1
,
ξ
2
,
ξ
3
{\displaystyle \xi ^{1},\xi ^{2},\xi ^{3}}
T
{\displaystyle {\boldsymbol {T}}}
[ 3]
∇
T
=
∂
T
∂
ξ
i
⊗
g
i
{\displaystyle {\boldsymbol {\nabla }}{\boldsymbol {T}}={\frac {\partial {\boldsymbol {T}}}{\partial \xi ^{i}}}\otimes \mathbf {g} ^{i}}
De esta definición se obtienen las siguientes relaciones para los gradientes de un campo escalar
ϕ
{\displaystyle \phi }
v y un campo tensorial de segundo orden
S
{\displaystyle {\boldsymbol {S}}}
∇
ϕ
=
∂
ϕ
∂
ξ
i
g
i
∇
v
=
∂
(
v
j
g
j
)
∂
ξ
i
⊗
g
i
=
(
∂
v
j
∂
ξ
i
+
v
k
Γ
i
k
j
)
g
j
⊗
g
i
=
(
∂
v
j
∂
ξ
i
−
v
k
Γ
i
j
k
)
g
j
⊗
g
i
∇
S
=
∂
(
S
j
k
g
j
⊗
g
k
)
∂
ξ
i
⊗
g
i
=
(
∂
S
j
k
∂
ξ
i
−
S
l
k
Γ
i
j
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−
S
j
l
Γ
i
k
l
)
g
j
⊗
g
k
⊗
g
i
{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\phi &={\frac {\partial \phi }{\partial \xi ^{i}}}~\mathbf {g} ^{i}\\{\boldsymbol {\nabla }}\mathbf {v} &={\frac {\partial \left(v^{j}\mathbf {g} _{j}\right)}{\partial \xi ^{i}}}\otimes \mathbf {g} ^{i}=\left({\frac {\partial v^{j}}{\partial \xi ^{i}}}+v^{k}~\Gamma _{ik}^{j}\right)~\mathbf {g} _{j}\otimes \mathbf {g} ^{i}=\left({\frac {\partial v_{j}}{\partial \xi ^{i}}}-v_{k}~\Gamma _{ij}^{k}\right)~\mathbf {g} ^{j}\otimes \mathbf {g} ^{i}\\{\boldsymbol {\nabla }}{\boldsymbol {S}}&={\frac {\partial \left(S_{jk}~\mathbf {g} ^{j}\otimes \mathbf {g} ^{k}\right)}{\partial \xi ^{i}}}\otimes \mathbf {g} ^{i}=\left({\frac {\partial S_{jk}}{\partial \xi _{i}}}-S_{lk}~\Gamma _{ij}^{l}-S_{jl}~\Gamma _{ik}^{l}\right)~\mathbf {g} ^{j}\otimes \mathbf {g} ^{k}\otimes \mathbf {g} ^{i}\end{aligned}}}
donde los símbolos de Christoffel
Γ
i
j
k
{\displaystyle \Gamma _{ij}^{k}}
Γ
i
j
k
g
k
=
∂
g
i
∂
ξ
j
⟹
Γ
i
j
k
=
∂
g
i
∂
ξ
j
⋅
g
k
=
−
g
i
⋅
∂
g
k
∂
ξ
j
{\displaystyle \Gamma _{ij}^{k}~\mathbf {g} _{k}={\frac {\partial \mathbf {g} _{i}}{\partial \xi ^{j}}}\quad \implies \quad \Gamma _{ij}^{k}={\frac {\partial \mathbf {g} _{i}}{\partial \xi ^{j}}}\cdot \mathbf {g} ^{k}=-\mathbf {g} _{i}\cdot {\frac {\partial \mathbf {g} ^{k}}{\partial \xi ^{j}}}}
editar
En coordenadas cilíndricas , el gradiente viene dado por
∇
ϕ
=
∂
ϕ
∂
r
e
r
+
1
r
∂
ϕ
∂
θ
e
θ
+
∂
ϕ
∂
z
e
z
∇
v
=
∂
v
r
∂
r
e
r
⊗
e
r
+
1
r
(
∂
v
r
∂
θ
−
v
θ
)
e
r
⊗
e
θ
+
∂
v
r
∂
z
e
r
⊗
e
z
+
∂
v
θ
∂
r
e
θ
⊗
e
r
+
1
r
(
∂
v
θ
∂
θ
+
v
r
)
e
θ
⊗
e
θ
+
∂
v
θ
∂
z
e
θ
⊗
e
z
+
∂
v
z
∂
r
e
z
⊗
e
r
+
1
r
∂
v
z
∂
θ
e
z
⊗
e
θ
+
∂
v
z
∂
z
e
z
⊗
e
z
∇
S
=
∂
S
r
r
∂
r
e
r
⊗
e
r
⊗
e
r
+
∂
S
r
r
∂
z
e
r
⊗
e
r
⊗
e
z
+
1
r
[
∂
S
r
r
∂
θ
−
(
S
θ
r
+
S
r
θ
)
]
e
r
⊗
e
r
⊗
e
θ
+
∂
S
r
θ
∂
r
e
r
⊗
e
θ
⊗
e
r
+
∂
S
r
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∂
z
e
r
⊗
e
θ
⊗
e
z
+
1
r
[
∂
S
r
θ
∂
θ
+
(
S
r
r
−
S
θ
θ
)
]
e
r
⊗
e
θ
⊗
e
θ
+
∂
S
r
z
∂
r
e
r
⊗
e
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⊗
e
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+
∂
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∂
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e
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e
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⊗
e
z
+
1
r
[
∂
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∂
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−
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θ
z
]
e
r
⊗
e
z
⊗
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+
∂
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∂
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+
1
r
[
∂
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+
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r
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−
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θ
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)
]
e
θ
⊗
e
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⊗
e
θ
+
∂
S
θ
θ
∂
r
e
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⊗
e
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+
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∂
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e
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e
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+
1
r
[
∂
S
θ
θ
∂
θ
+
(
S
r
θ
+
S
θ
r
)
]
e
θ
⊗
e
θ
⊗
e
θ
+
∂
S
θ
z
∂
r
e
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⊗
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z
⊗
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r
+
∂
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∂
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e
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⊗
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⊗
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+
1
r
[
∂
S
θ
z
∂
θ
+
S
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z
]
e
θ
⊗
e
z
⊗
e
θ
+
∂
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+
∂
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e
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⊗
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+
1
r
[
∂
S
z
r
∂
θ
−
S
z
θ
]
e
z
⊗
e
r
⊗
e
θ
+
∂
S
z
θ
∂
r
e
z
⊗
e
θ
⊗
e
r
+
∂
S
z
θ
∂
z
e
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e
θ
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+
1
r
[
∂
S
z
θ
∂
θ
+
S
z
r
]
e
z
⊗
e
θ
⊗
e
θ
+
∂
S
z
z
∂
r
e
z
⊗
e
z
⊗
e
r
+
∂
S
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∂
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e
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+
1
r
∂
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θ
e
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⊗
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θ
{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\phi ={}\quad &{\frac {\partial \phi }{\partial r}}~\mathbf {e} _{r}+{\frac {1}{r}}~{\frac {\partial \phi }{\partial \theta }}~\mathbf {e} _{\theta }+{\frac {\partial \phi }{\partial z}}~\mathbf {e} _{z}\\{\boldsymbol {\nabla }}\mathbf {v} ={}\quad &{\frac {\partial v_{r}}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\frac {1}{r}}\left({\frac {\partial v_{r}}{\partial \theta }}-v_{\theta }\right)~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }+{\frac {\partial v_{r}}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\\{}+{}&{\frac {\partial v_{\theta }}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\frac {1}{r}}\left({\frac {\partial v_{\theta }}{\partial \theta }}+v_{r}\right)~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }+{\frac {\partial v_{\theta }}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\\{}+{}&{\frac {\partial v_{z}}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\frac {1}{r}}{\frac {\partial v_{z}}{\partial \theta }}~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }+{\frac {\partial v_{z}}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\\{\boldsymbol {\nabla }}{\boldsymbol {S}}={}\quad &{\frac {\partial S_{rr}}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\frac {\partial S_{rr}}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{z}+{\frac {1}{r}}\left[{\frac {\partial S_{rr}}{\partial \theta }}-(S_{\theta r}+S_{r\theta })\right]~\mathbf {e} _{r}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\\{}+{}&{\frac {\partial S_{r\theta }}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\frac {\partial S_{r\theta }}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{z}+{\frac {1}{r}}\left[{\frac {\partial S_{r\theta }}{\partial \theta }}+(S_{rr}-S_{\theta \theta })\right]~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\\{}+{}&{\frac {\partial S_{rz}}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\frac {\partial S_{rz}}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{z}+{\frac {1}{r}}\left[{\frac {\partial S_{rz}}{\partial \theta }}-S_{\theta z}\right]~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\\{}+{}&{\frac {\partial S_{\theta r}}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\frac {\partial S_{\theta r}}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{z}+{\frac {1}{r}}\left[{\frac {\partial S_{\theta r}}{\partial \theta }}+(S_{rr}-S_{\theta \theta })\right]~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\\{}+{}&{\frac {\partial S_{\theta \theta }}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\frac {\partial S_{\theta \theta }}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{z}+{\frac {1}{r}}\left[{\frac {\partial S_{\theta \theta }}{\partial \theta }}+(S_{r\theta }+S_{\theta r})\right]~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\\{}+{}&{\frac {\partial S_{\theta z}}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\frac {\partial S_{\theta z}}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{z}+{\frac {1}{r}}\left[{\frac {\partial S_{\theta z}}{\partial \theta }}+S_{rz}\right]~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\\{}+{}&{\frac {\partial S_{zr}}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\frac {\partial S_{zr}}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{z}+{\frac {1}{r}}\left[{\frac {\partial S_{zr}}{\partial \theta }}-S_{z\theta }\right]~\mathbf {e} _{z}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\\{}+{}&{\frac {\partial S_{z\theta }}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\frac {\partial S_{z\theta }}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{z}+{\frac {1}{r}}\left[{\frac {\partial S_{z\theta }}{\partial \theta }}+S_{zr}\right]~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\\{}+{}&{\frac {\partial S_{zz}}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\frac {\partial S_{zz}}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{z}+{\frac {1}{r}}~{\frac {\partial S_{zz}}{\partial \theta }}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\end{aligned}}}
Divergencia de un campo tensorial
editar
La divergencia de un campo tensorial
T
(
x
)
{\displaystyle {\boldsymbol {T}}(\mathbf {x} )}
(
∇
⋅
T
)
⋅
c
=
∇
⋅
(
c
⋅
T
T
)
;
∇
⋅
v
=
tr
(
∇
v
)
{\displaystyle ({\boldsymbol {\nabla }}\cdot {\boldsymbol {T}})\cdot \mathbf {c} ={\boldsymbol {\nabla }}\cdot \left(\mathbf {c} \cdot {\boldsymbol {T}}^{\textsf {T}}\right)~;\qquad {\boldsymbol {\nabla }}\cdot \mathbf {v} ={\text{tr}}({\boldsymbol {\nabla }}\mathbf {v} )}
donde c es un vector constante arbitrario y v es un campo vectorial. Si
T
{\displaystyle {\boldsymbol {T}}}
n > 1, entonces la divergencia del campo es un tensor de orden n - 1.
Coordenadas cartesianas
editar
En un sistema de coordenadas cartesiano se tienen las siguientes relaciones para un campo vectorial v' y un campo tensorial de segundo orden
S
{\displaystyle {\boldsymbol {S}}}
∇
⋅
v
=
∂
v
i
∂
x
i
=
v
i
,
i
∇
⋅
S
=
∂
S
i
k
∂
x
i
e
k
=
S
i
k
,
i
e
k
{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\cdot \mathbf {v} &={\frac {\partial v_{i}}{\partial x_{i}}}=v_{i,i}\\{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}&={\frac {\partial S_{ik}}{\partial x_{i}}}~\mathbf {e} _{k}=S_{ik,i}~\mathbf {e} _{k}\end{aligned}}}
donde con la notación tensorial indexada para derivadas parciales se utiliza en las expresiones situadas más a la derecha. Tenga en cuenta que
∇
⋅
S
≠
∇
⋅
S
T
.
{\displaystyle {\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}\neq {\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}^{\textsf {T}}.}
Para un tensor simétrico de segundo orden, la divergencia también suele escribirse como[ 4]
∇
⋅
S
=
∂
S
k
i
∂
x
i
e
k
=
S
k
i
,
i
e
k
{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}&={\cfrac {\partial S_{ki}}{\partial x_{i}}}~\mathbf {e} _{k}=S_{ki,i}~\mathbf {e} _{k}\end{aligned}}}
La expresión anterior se utiliza a veces como definición de
∇
⋅
S
{\displaystyle {\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}}
div
S
{\displaystyle \operatorname {div} {\boldsymbol {S}}}
La diferencia surge de si la diferenciación se realiza respecto de las filas o columnas de
S
{\displaystyle {\boldsymbol {S}}}
S
{\displaystyle \mathbf {S} }
v
{\displaystyle \mathbf {v} }
∇
⋅
(
∇
v
)
=
∇
⋅
(
v
i
,
j
e
i
⊗
e
j
)
=
v
i
,
j
i
e
i
⋅
e
i
⊗
e
j
=
(
∇
⋅
v
)
,
j
e
j
=
∇
(
∇
⋅
v
)
∇
⋅
[
(
∇
v
)
T
]
=
∇
⋅
(
v
j
,
i
e
i
⊗
e
j
)
=
v
j
,
i
i
e
i
⋅
e
i
⊗
e
j
=
∇
2
v
j
e
j
=
∇
2
v
{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\cdot \left({\boldsymbol {\nabla }}\mathbf {v} \right)&={\boldsymbol {\nabla }}\cdot \left(v_{i,j}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\right)=v_{i,ji}~\mathbf {e} _{i}\cdot \mathbf {e} _{i}\otimes \mathbf {e} _{j}=\left({\boldsymbol {\nabla }}\cdot \mathbf {v} \right)_{,j}~\mathbf {e} _{j}={\boldsymbol {\nabla }}\left({\boldsymbol {\nabla }}\cdot \mathbf {v} \right)\\{\boldsymbol {\nabla }}\cdot \left[\left({\boldsymbol {\nabla }}\mathbf {v} \right)^{\textsf {T}}\right]&={\boldsymbol {\nabla }}\cdot \left(v_{j,i}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\right)=v_{j,ii}~\mathbf {e} _{i}\cdot \mathbf {e} _{i}\otimes \mathbf {e} _{j}={\boldsymbol {\nabla }}^{2}v_{j}~\mathbf {e} _{j}={\boldsymbol {\nabla }}^{2}\mathbf {v} \end{aligned}}}
La última ecuación es equivalente a la definición/interpretación alternativa[ 4]
(
∇
⋅
)
alt
(
∇
v
)
=
(
∇
⋅
)
alt
(
v
i
,
j
e
i
⊗
e
j
)
=
v
i
,
j
j
e
i
⊗
e
j
⋅
e
j
=
∇
2
v
i
e
i
=
∇
2
v
{\displaystyle {\begin{aligned}\left({\boldsymbol {\nabla }}\cdot \right)_{\text{alt}}\left({\boldsymbol {\nabla }}\mathbf {v} \right)=\left({\boldsymbol {\nabla }}\cdot \right)_{\text{alt}}\left(v_{i,j}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\right)=v_{i,jj}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\cdot \mathbf {e} _{j}={\boldsymbol {\nabla }}^{2}v_{i}~\mathbf {e} _{i}={\boldsymbol {\nabla }}^{2}\mathbf {v} \end{aligned}}}
editar
En coordenadas curvilíneas, las divergencias de un campo vectorial v' y de un campo tensorial de segundo orden
S
{\displaystyle {\boldsymbol {S}}}
∇
⋅
v
=
(
∂
v
i
∂
ξ
i
+
v
k
Γ
i
k
i
)
∇
⋅
S
=
(
∂
S
i
k
∂
ξ
i
−
S
l
k
Γ
i
i
l
−
S
i
l
Γ
i
k
l
)
g
k
{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\cdot \mathbf {v} &=\left({\cfrac {\partial v^{i}}{\partial \xi ^{i}}}+v^{k}~\Gamma _{ik}^{i}\right)\\{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}&=\left({\cfrac {\partial S_{ik}}{\partial \xi _{i}}}-S_{lk}~\Gamma _{ii}^{l}-S_{il}~\Gamma _{ik}^{l}\right)~\mathbf {g} ^{k}\end{aligned}}}
Más generalmente,
∇
⋅
S
=
[
∂
S
i
j
∂
q
k
−
Γ
k
i
l
S
l
j
−
Γ
k
j
l
S
i
l
]
g
i
k
b
j
=
[
∂
S
i
j
∂
q
i
+
Γ
i
l
i
S
l
j
+
Γ
i
l
j
S
i
l
]
b
j
=
[
∂
S
j
i
∂
q
i
+
Γ
i
l
i
S
j
l
−
Γ
i
j
l
S
l
i
]
b
j
=
[
∂
S
i
j
∂
q
k
−
Γ
i
k
l
S
l
j
+
Γ
k
l
j
S
i
l
]
g
i
k
b
j
{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}&=\left[{\cfrac {\partial S_{ij}}{\partial q^{k}}}-\Gamma _{ki}^{l}~S_{lj}-\Gamma _{kj}^{l}~S_{il}\right]~g^{ik}~\mathbf {b} ^{j}\\[8pt]&=\left[{\cfrac {\partial S^{ij}}{\partial q^{i}}}+\Gamma _{il}^{i}~S^{lj}+\Gamma _{il}^{j}~S^{il}\right]~\mathbf {b} _{j}\\[8pt]&=\left[{\cfrac {\partial S_{~j}^{i}}{\partial q^{i}}}+\Gamma _{il}^{i}~S_{~j}^{l}-\Gamma _{ij}^{l}~S_{~l}^{i}\right]~\mathbf {b} ^{j}\\[8pt]&=\left[{\cfrac {\partial S_{i}^{~j}}{\partial q^{k}}}-\Gamma _{ik}^{l}~S_{l}^{~j}+\Gamma _{kl}^{j}~S_{i}^{~l}\right]~g^{ik}~\mathbf {b} _{j}\end{aligned}}}
editar
En coordenadas polares cilíndricas
∇
⋅
v
=
∂
v
r
∂
r
+
1
r
(
∂
v
θ
∂
θ
+
v
r
)
+
∂
v
z
∂
z
∇
⋅
S
=
∂
S
r
r
∂
r
e
r
+
∂
S
r
θ
∂
r
e
θ
+
∂
S
r
z
∂
r
e
z
+
1
r
[
∂
S
θ
r
∂
θ
+
(
S
r
r
−
S
θ
θ
)
]
e
r
+
1
r
[
∂
S
θ
θ
∂
θ
+
(
S
r
θ
+
S
θ
r
)
]
e
θ
+
1
r
[
∂
S
θ
z
∂
θ
+
S
r
z
]
e
z
+
∂
S
z
r
∂
z
e
r
+
∂
S
z
θ
∂
z
e
θ
+
∂
S
z
z
∂
z
e
z
{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\cdot \mathbf {v} =\quad &{\frac {\partial v_{r}}{\partial r}}+{\frac {1}{r}}\left({\frac {\partial v_{\theta }}{\partial \theta }}+v_{r}\right)+{\frac {\partial v_{z}}{\partial z}}\\{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}=\quad &{\frac {\partial S_{rr}}{\partial r}}~\mathbf {e} _{r}+{\frac {\partial S_{r\theta }}{\partial r}}~\mathbf {e} _{\theta }+{\frac {\partial S_{rz}}{\partial r}}~\mathbf {e} _{z}\\{}+{}&{\frac {1}{r}}\left[{\frac {\partial S_{\theta r}}{\partial \theta }}+(S_{rr}-S_{\theta \theta })\right]~\mathbf {e} _{r}+{\frac {1}{r}}\left[{\frac {\partial S_{\theta \theta }}{\partial \theta }}+(S_{r\theta }+S_{\theta r})\right]~\mathbf {e} _{\theta }+{\frac {1}{r}}\left[{\frac {\partial S_{\theta z}}{\partial \theta }}+S_{rz}\right]~\mathbf {e} _{z}\\{}+{}&{\frac {\partial S_{zr}}{\partial z}}~\mathbf {e} _{r}+{\frac {\partial S_{z\theta }}{\partial z}}~\mathbf {e} _{\theta }+{\frac {\partial S_{zz}}{\partial z}}~\mathbf {e} _{z}\end{aligned}}}
Rotacional de un campo tensorial
editar
El rotacional de un campo tensorial de orden n > 1
T
(
x
)
{\displaystyle {\boldsymbol {T}}(\mathbf {x} )}
(
∇
×
T
)
⋅
c
=
∇
×
(
c
⋅
T
)
;
(
∇
×
v
)
⋅
c
=
∇
⋅
(
v
×
c
)
{\displaystyle ({\boldsymbol {\nabla }}\times {\boldsymbol {T}})\cdot \mathbf {c} ={\boldsymbol {\nabla }}\times (\mathbf {c} \cdot {\boldsymbol {T}})~;\qquad ({\boldsymbol {\nabla }}\times \mathbf {v} )\cdot \mathbf {c} ={\boldsymbol {\nabla }}\cdot (\mathbf {v} \times \mathbf {c} )}
donde c es un vector constante arbitrario y v es un campo vectorial.
editar
Considérese un campo vectorial v y un vector constante arbitrario c . En notación indexada, el producto cruzado viene dado por
v
×
c
=
ε
i
j
k
v
j
c
k
e
i
{\displaystyle \mathbf {v} \times \mathbf {c} =\varepsilon _{ijk}~v_{j}~c_{k}~\mathbf {e} _{i}}
donde
ε
i
j
k
{\displaystyle \varepsilon _{ijk}}
símbolo de permutación , también conocido como símbolo de Levi-Civita. Entonces,
∇
⋅
(
v
×
c
)
=
ε
i
j
k
v
j
,
i
c
k
=
(
ε
i
j
k
v
j
,
i
e
k
)
⋅
c
=
(
∇
×
v
)
⋅
c
{\displaystyle {\boldsymbol {\nabla }}\cdot (\mathbf {v} \times \mathbf {c} )=\varepsilon _{ijk}~v_{j,i}~c_{k}=(\varepsilon _{ijk}~v_{j,i}~\mathbf {e} _{k})\cdot \mathbf {c} =({\boldsymbol {\nabla }}\times \mathbf {v} )\cdot \mathbf {c} }
Por lo tanto,
∇
×
v
=
ε
i
j
k
v
j
,
i
e
k
{\displaystyle {\boldsymbol {\nabla }}\times \mathbf {v} =\varepsilon _{ijk}~v_{j,i}~\mathbf {e} _{k}}
Rotacional de un campo tensorial de segundo orden
editar
Para un tensor de segundo orden
S
{\displaystyle {\boldsymbol {S}}}
c
⋅
S
=
c
m
S
m
j
e
j
{\displaystyle \mathbf {c} \cdot {\boldsymbol {S}}=c_{m}~S_{mj}~\mathbf {e} _{j}}
Por tanto, utilizando la definición de la curvatura de un campo tensorial de primer orden,
∇
×
(
c
⋅
S
)
=
ε
i
j
k
c
m
S
m
j
,
i
e
k
=
(
ε
i
j
k
S
m
j
,
i
e
k
⊗
e
m
)
⋅
c
=
(
∇
×
S
)
⋅
c
{\displaystyle {\boldsymbol {\nabla }}\times (\mathbf {c} \cdot {\boldsymbol {S}})=\varepsilon _{ijk}~c_{m}~S_{mj,i}~\mathbf {e} _{k}=(\varepsilon _{ijk}~S_{mj,i}~\mathbf {e} _{k}\otimes \mathbf {e} _{m})\cdot \mathbf {c} =({\boldsymbol {\nabla }}\times {\boldsymbol {S}})\cdot \mathbf {c} }
Por lo tanto, se tiene que
∇
×
S
=
ε
i
j
k
S
m
j
,
i
e
k
⊗
e
m
{\displaystyle {\boldsymbol {\nabla }}\times {\boldsymbol {S}}=\varepsilon _{ijk}~S_{mj,i}~\mathbf {e} _{k}\otimes \mathbf {e} _{m}}
Identidades que involucran la curvatura de un campo tensorial
editar
La identidad más comúnmente utilizada que involucra la curvatura de un campo tensorial,
T
{\displaystyle {\boldsymbol {T}}}
∇
×
(
∇
T
)
=
0
{\displaystyle {\boldsymbol {\nabla }}\times ({\boldsymbol {\nabla }}{\boldsymbol {T}})={\boldsymbol {0}}}
Esta identidad es válida para campos tensoriales de todos los órdenes. Para el caso importante de un tensor de segundo orden,
S
{\displaystyle {\boldsymbol {S}}}
∇
×
(
∇
S
)
=
0
⟹
S
m
i
,
j
−
S
m
j
,
i
=
0
{\displaystyle {\boldsymbol {\nabla }}\times ({\boldsymbol {\nabla }}{\boldsymbol {S}})={\boldsymbol {0}}\quad \implies \quad S_{mi,j}-S_{mj,i}=0}
Derivada del determinante de un tensor de segundo orden
editar
La derivada del determinante de un tensor de segundo orden
A
{\displaystyle {\boldsymbol {A}}}
∂
∂
A
det
(
A
)
=
det
(
A
)
[
A
−
1
]
T
.
{\displaystyle {\frac {\partial }{\partial {\boldsymbol {A}}}}\det({\boldsymbol {A}})=\det({\boldsymbol {A}})~\left[{\boldsymbol {A}}^{-1}\right]^{\textsf {T}}~.}
En términos ortonormales, las componentes de
A
{\displaystyle {\boldsymbol {A}}}
A . En ese caso, el lado derecho corresponde a los cofactores de la matriz.
Derivadas de los invariantes de un tensor de segundo orden
editar
Los principales invariantes de un tensor de segundo orden son
I
1
(
A
)
=
tr
A
I
2
(
A
)
=
1
2
[
(
tr
A
)
2
−
tr
A
2
]
I
3
(
A
)
=
det
(
A
)
{\displaystyle {\begin{aligned}I_{1}({\boldsymbol {A}})&={\text{tr}}{\boldsymbol {A}}\\I_{2}({\boldsymbol {A}})&={\frac {1}{2}}\left[({\text{tr}}{\boldsymbol {A}})^{2}-{\text{tr}}{{\boldsymbol {A}}^{2}}\right]\\I_{3}({\boldsymbol {A}})&=\det({\boldsymbol {A}})\end{aligned}}}
Las derivadas de estos tres invariantes con respecto a
A
{\displaystyle {\boldsymbol {A}}}
∂
I
1
∂
A
=
1
∂
I
2
∂
A
=
I
1
1
−
A
T
∂
I
3
∂
A
=
det
(
A
)
[
A
−
1
]
T
=
I
2
1
−
A
T
(
I
1
1
−
A
T
)
=
(
A
2
−
I
1
A
+
I
2
1
)
T
{\displaystyle {\begin{aligned}{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}&={\boldsymbol {\mathit {1}}}\\[3pt]{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}&=I_{1}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}\\[3pt]{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}&=\det({\boldsymbol {A}})~\left[{\boldsymbol {A}}^{-1}\right]^{\textsf {T}}=I_{2}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}~\left(I_{1}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}\right)=\left({\boldsymbol {A}}^{2}-I_{1}~{\boldsymbol {A}}+I_{2}~{\boldsymbol {\mathit {1}}}\right)^{\textsf {T}}\end{aligned}}}
Demostración
De la derivada del determinante se sabe que
∂
I
3
∂
A
=
(
A
)
[
A
−
1
]
T
.
{\displaystyle {\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}=(A)~[A^{-}1]^{T}~.}
Para las derivadas de las otras dos invariantes, se retoma la ecuación característica
det
(
λ
1
+
A
)
=
λ
3
+
I
1
(
A
)
λ
2
+
I
2
(
A
)
λ
+
I
3
(
A
)
.
{\displaystyle \det(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})=\lambda ^{3}+I_{1}({\boldsymbol {A}})~\lambda ^{2}+I_{2}({\boldsymbol {A}})~\lambda +I_{3}({\boldsymbol {A}})~.}
Utilizando el mismo enfoque que para el determinante de un tensor, se puede demostrar que
∂
∂
A
det
(
λ
1
+
A
)
=
det
(
λ
1
+
A
)
[
(
λ
1
+
A
)
−
1
]
T
.
{\displaystyle {\frac {\partial }{\partial {\boldsymbol {A}}}}\det(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})=\det(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})~\left[(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})^{-1}\right]^{\textsf {T}}~.}
Ahora, el lado izquierdo se puede expandir como
∂
∂
A
det
(
λ
1
+
A
)
=
∂
∂
A
[
λ
3
+
I
1
(
A
)
λ
2
+
I
2
(
A
)
λ
+
I
3
(
A
)
]
=
∂
I
1
∂
A
λ
2
+
∂
I
2
∂
A
λ
+
∂
I
3
∂
A
.
{\displaystyle {\begin{aligned}{\frac {\partial }{\partial {\boldsymbol {A}}}}\det(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})&={\frac {\partial }{\partial {\boldsymbol {A}}}}\left[\lambda ^{3}+I_{1}({\boldsymbol {A}})~\lambda ^{2}+I_{2}({\boldsymbol {A}})~\lambda +I_{3}({\boldsymbol {A}})\right]\\&={\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda +{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}~.\end{aligned}}}
Por eso
∂
I
1
∂
A
λ
2
+
∂
I
2
∂
A
λ
+
∂
I
3
∂
A
=
det
(
λ
1
+
A
)
[
(
λ
1
+
A
)
−
1
]
T
{\displaystyle {\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda +{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}=\det(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})~\left[(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})^{-1}\right]^{\textsf {T}}}
o,
(
λ
1
+
A
)
T
⋅
[
∂
I
1
∂
A
λ
2
+
∂
I
2
∂
A
λ
+
∂
I
3
∂
A
]
=
det
(
λ
1
+
A
)
1
.
{\displaystyle (\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})^{\textsf {T}}\cdot \left[{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda +{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}\right]=\det(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})~{\boldsymbol {\mathit {1}}}~.}
Expandir el lado derecho y separar términos en el lado izquierdo da
(
λ
1
+
A
T
)
⋅
[
∂
I
1
∂
A
λ
2
+
∂
I
2
∂
A
λ
+
∂
I
3
∂
A
]
=
[
λ
3
+
I
1
λ
2
+
I
2
λ
+
I
3
]
1
{\displaystyle \left(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}}^{\textsf {T}}\right)\cdot \left[{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda +{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}\right]=\left[\lambda ^{3}+I_{1}~\lambda ^{2}+I_{2}~\lambda +I_{3}\right]{\boldsymbol {\mathit {1}}}}
o,
[
∂
I
1
∂
A
λ
3
+
∂
I
2
∂
A
λ
2
+
∂
I
3
∂
A
λ
]
1
+
A
T
⋅
∂
I
1
∂
A
λ
2
+
A
T
⋅
∂
I
2
∂
A
λ
+
A
T
⋅
∂
I
3
∂
A
=
[
λ
3
+
I
1
λ
2
+
I
2
λ
+
I
3
]
1
.
{\displaystyle {\begin{aligned}\left[{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{3}\right.&\left.+{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}~\lambda \right]{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda +{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}\\&=\left[\lambda ^{3}+I_{1}~\lambda ^{2}+I_{2}~\lambda +I_{3}\right]{\boldsymbol {\mathit {1}}}~.\end{aligned}}}
Si se define
I
0
:=
1
{\displaystyle I_{0}:=1}
I
4
:=
0
{\displaystyle I_{4}:=0}
[
∂
I
1
∂
A
λ
3
+
∂
I
2
∂
A
λ
2
+
∂
I
3
∂
A
λ
+
∂
I
4
∂
A
]
1
+
A
T
⋅
∂
I
0
∂
A
λ
3
+
A
T
⋅
∂
I
1
∂
A
λ
2
+
A
T
⋅
∂
I
2
∂
A
λ
+
A
T
⋅
∂
I
3
∂
A
=
[
I
0
λ
3
+
I
1
λ
2
+
I
2
λ
+
I
3
]
1
.
{\displaystyle {\begin{aligned}\left[{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{3}\right.&\left.+{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}~\lambda +{\frac {\partial I_{4}}{\partial {\boldsymbol {A}}}}\right]{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{0}}{\partial {\boldsymbol {A}}}}~\lambda ^{3}+{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda +{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}\\&=\left[I_{0}~\lambda ^{3}+I_{1}~\lambda ^{2}+I_{2}~\lambda +I_{3}\right]{\boldsymbol {\mathit {1}}}~.\end{aligned}}}
Reuniendo términos que contienen varias potencias de λ, se obtiene
λ
3
(
I
0
1
−
∂
I
1
∂
A
1
−
A
T
⋅
∂
I
0
∂
A
)
+
λ
2
(
I
1
1
−
∂
I
2
∂
A
1
−
A
T
⋅
∂
I
1
∂
A
)
+
λ
(
I
2
1
−
∂
I
3
∂
A
1
−
A
T
⋅
∂
I
2
∂
A
)
+
(
I
3
1
−
∂
I
4
∂
A
1
−
A
T
⋅
∂
I
3
∂
A
)
=
0
.
{\displaystyle {\begin{aligned}\lambda ^{3}&\left(I_{0}~{\boldsymbol {\mathit {1}}}-{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{0}}{\partial {\boldsymbol {A}}}}\right)+\lambda ^{2}\left(I_{1}~{\boldsymbol {\mathit {1}}}-{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}\right)+\\&\qquad \qquad \lambda \left(I_{2}~{\boldsymbol {\mathit {1}}}-{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}\right)+\left(I_{3}~{\boldsymbol {\mathit {1}}}-{\frac {\partial I_{4}}{\partial {\boldsymbol {A}}}}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}\right)=0~.\end{aligned}}}
Entonces, invocando la arbitrariedad de λ, se tiene que
I
0
1
−
∂
I
1
∂
A
1
−
A
T
⋅
∂
I
0
∂
A
=
0
I
1
1
−
∂
I
2
∂
A
1
−
I
2
1
−
∂
I
3
∂
A
1
−
A
T
⋅
∂
I
2
∂
A
=
0
I
3
1
−
∂
I
4
∂
A
1
−
A
T
⋅
∂
I
3
∂
A
=
0
.
{\displaystyle {\begin{aligned}I_{0}~{\boldsymbol {\mathit {1}}}-{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{0}}{\partial {\boldsymbol {A}}}}&=0\\I_{1}~{\boldsymbol {\mathit {1}}}-{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~{\boldsymbol {\mathit {1}}}-I_{2}~{\boldsymbol {\mathit {1}}}-{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}&=0\\I_{3}~{\boldsymbol {\mathit {1}}}-{\frac {\partial I_{4}}{\partial {\boldsymbol {A}}}}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}&=0~.\end{aligned}}}
Esto implica que
∂
I
1
∂
A
=
1
∂
I
2
∂
A
=
I
1
1
−
A
T
∂
I
3
∂
A
=
I
2
1
−
A
T
(
I
1
1
−
A
T
)
=
(
A
2
−
I
1
A
+
I
2
1
)
T
{\displaystyle {\begin{aligned}{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}&={\boldsymbol {\mathit {1}}}\\{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}&=I_{1}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}\\{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}&=I_{2}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}~\left(I_{1}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}\right)=\left({\boldsymbol {A}}^{2}-I_{1}~{\boldsymbol {A}}+I_{2}~{\boldsymbol {\mathit {1}}}\right)^{\textsf {T}}\end{aligned}}}
Derivada del tensor de identidad de segundo orden
editar
editar
Sea
A
{\displaystyle {\boldsymbol {A}}}
∂
A
∂
A
:
T
=
[
∂
∂
α
(
A
+
α
T
)
]
α
=
0
=
T
=
I
:
T
{\displaystyle {\frac {\partial {\boldsymbol {A}}}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}=\left[{\frac {\partial }{\partial \alpha }}({\boldsymbol {A}}+\alpha ~{\boldsymbol {T}})\right]_{\alpha =0}={\boldsymbol {T}}={\boldsymbol {\mathsf {I}}}:{\boldsymbol {T}}}
Por lo tanto,
∂
A
∂
A
=
I
{\displaystyle {\frac {\partial {\boldsymbol {A}}}{\partial {\boldsymbol {A}}}}={\boldsymbol {\mathsf {I}}}}
Aquí
I
{\displaystyle {\boldsymbol {\mathsf {I}}}}
I
=
δ
i
k
δ
j
l
e
i
⊗
e
j
⊗
e
k
⊗
e
l
{\displaystyle {\boldsymbol {\mathsf {I}}}=\delta _{ik}~\delta _{jl}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\otimes \mathbf {e} _{k}\otimes \mathbf {e} _{l}}
Este resultado implica que
∂
A
T
∂
A
:
T
=
I
T
:
T
=
T
T
{\displaystyle {\frac {\partial {\boldsymbol {A}}^{\textsf {T}}}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}={\boldsymbol {\mathsf {I}}}^{\textsf {T}}:{\boldsymbol {T}}={\boldsymbol {T}}^{\textsf {T}}}
donde
I
T
=
δ
j
k
δ
i
l
e
i
⊗
e
j
⊗
e
k
⊗
e
l
{\displaystyle {\boldsymbol {\mathsf {I}}}^{\textsf {T}}=\delta _{jk}~\delta _{il}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\otimes \mathbf {e} _{k}\otimes \mathbf {e} _{l}}
Por lo tanto, si el tensor
A
{\displaystyle {\boldsymbol {A}}}
∂
A
∂
A
=
I
(
s
)
=
1
2
(
I
+
I
T
)
{\displaystyle {\frac {\partial {\boldsymbol {A}}}{\partial {\boldsymbol {A}}}}={\boldsymbol {\mathsf {I}}}^{(s)}={\frac {1}{2}}~\left({\boldsymbol {\mathsf {I}}}+{\boldsymbol {\mathsf {I}}}^{\textsf {T}}\right)}
donde el tensor de identidad simétrico de cuarto orden es
I
(
s
)
=
1
2
(
δ
i
k
δ
j
l
+
δ
i
l
δ
j
k
)
e
i
⊗
e
j
⊗
e
k
⊗
e
l
{\displaystyle {\boldsymbol {\mathsf {I}}}^{(s)}={\frac {1}{2}}~(\delta _{ik}~\delta _{jl}+\delta _{il}~\delta _{jk})~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\otimes \mathbf {e} _{k}\otimes \mathbf {e} _{l}}
Derivada del inverso de un tensor de segundo orden
editar
Dominio
Ω
{\displaystyle \Omega }
Γ
{\displaystyle \Gamma }
n
{\displaystyle \mathbf {n} }
Otra operación importante relacionada con las derivadas tensoriales en la mecánica continua es la integración por partes. La fórmula de integración por partes se puede escribir como
∫
Ω
F
⊗
∇
G
d
Ω
=
∫
Γ
n
⊗
(
F
⊗
G
)
d
Γ
−
∫
Ω
G
⊗
∇
F
d
Ω
{\displaystyle \int _{\Omega }{\boldsymbol {F}}\otimes {\boldsymbol {\nabla }}{\boldsymbol {G}}\,d\Omega =\int _{\Gamma }\mathbf {n} \otimes ({\boldsymbol {F}}\otimes {\boldsymbol {G}})\,d\Gamma -\int _{\Omega }{\boldsymbol {G}}\otimes {\boldsymbol {\nabla }}{\boldsymbol {F}}\,d\Omega }
donde
F
{\displaystyle {\boldsymbol {F}}}
G
{\displaystyle {\boldsymbol {G}}}
n
{\displaystyle \mathbf {n} }
⊗
{\displaystyle \otimes }
∇
{\displaystyle {\boldsymbol {\nabla }}}
F
{\displaystyle {\boldsymbol {F}}}
teorema de la divergencia
∫
Ω
∇
G
d
Ω
=
∫
Γ
n
⊗
G
d
Γ
.
{\displaystyle \int _{\Omega }{\boldsymbol {\nabla }}{\boldsymbol {G}}\,d\Omega =\int _{\Gamma }\mathbf {n} \otimes {\boldsymbol {G}}\,d\Gamma \,.}
Se puede expresar la fórmula de integración por partes en coordenadas cartesianas con notación indexada como
∫
Ω
F
i
j
k
.
.
.
.
G
l
m
n
.
.
.
,
p
d
Ω
=
∫
Γ
n
p
F
i
j
k
.
.
.
G
l
m
n
.
.
.
d
Γ
−
∫
Ω
G
l
m
n
.
.
.
F
i
j
k
.
.
.
,
p
d
Ω
.
{\displaystyle \int _{\Omega }F_{ijk....}\,G_{lmn...,p}\,d\Omega =\int _{\Gamma }n_{p}\,F_{ijk...}\,G_{lmn...}\,d\Gamma -\int _{\Omega }G_{lmn...}\,F_{ijk...,p}\,d\Omega \,.}
Para el caso especial donde la operación del producto tensorial es una contracción de un índice y la operación del gradiente es una divergencia, y tanto
F
{\displaystyle {\boldsymbol {F}}}
G
{\displaystyle {\boldsymbol {G}}}
∫
Ω
F
⋅
(
∇
⋅
G
)
d
Ω
=
∫
Γ
n
⋅
(
G
⋅
F
T
)
d
Γ
−
∫
Ω
(
∇
F
)
:
G
T
d
Ω
.
{\displaystyle \int _{\Omega }{\boldsymbol {F}}\cdot ({\boldsymbol {\nabla }}\cdot {\boldsymbol {G}})\,d\Omega =\int _{\Gamma }\mathbf {n} \cdot \left({\boldsymbol {G}}\cdot {\boldsymbol {F}}^{\textsf {T}}\right)\,d\Gamma -\int _{\Omega }({\boldsymbol {\nabla }}{\boldsymbol {F}}):{\boldsymbol {G}}^{\textsf {T}}\,d\Omega \,.}
En notación indexada,
∫
Ω
F
i
j
G
p
j
,
p
d
Ω
=
∫
Γ
n
p
F
i
j
G
p
j
d
Γ
−
∫
Ω
G
p
j
F
i
j
,
p
d
Ω
.
{\displaystyle \int _{\Omega }F_{ij}\,G_{pj,p}\,d\Omega =\int _{\Gamma }n_{p}\,F_{ij}\,G_{pj}\,d\Gamma -\int _{\Omega }G_{pj}\,F_{ij,p}\,d\Omega \,.}
↑ J. C. Simo and T. J. R. Hughes, 1998, Computational Inelasticity , Springer
↑ J. E. Marsden and T. J. R. Hughes, 2000, Mathematical Foundations of Elasticity , Dover.
↑ R. W. Ogden, 2000, Nonlinear Elastic Deformations , Dover.
↑ a b Hjelmstad, Keith (2004). Fundamentals of Structural Mechanics . Springer Science & Business Media. p. 45. ISBN 9780387233307 
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/308eadd1b18b96a60ef33e8df365bccc97f0faea)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8835d592f0ae810b9b57fa81f345a4850f1f1ce)
![{\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/3916392ebeebca9ebddb31e76edb1c9f8c08b586)
![{\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/e2072ee23ca460f6f32851f397b5879a47033c3b)
![{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}{\boldsymbol {T}}\cdot \mathbf {c} &=\left.{\cfrac {d}{d\alpha }}~{\boldsymbol {T}}(x_{1}+\alpha c_{1},x_{2}+\alpha c_{2},x_{3}+\alpha c_{3})\right|_{\alpha =0}\equiv \left.{\cfrac {d}{d\alpha }}~{\boldsymbol {T}}(y_{1},y_{2},y_{3})\right|_{\alpha =0}\\&=\left[{\cfrac {\partial {\boldsymbol {T}}}{\partial y_{1}}}~{\cfrac {\partial y_{1}}{\partial \alpha }}+{\cfrac {\partial {\boldsymbol {T}}}{\partial y_{2}}}~{\cfrac {\partial y_{2}}{\partial \alpha }}+{\cfrac {\partial {\boldsymbol {T}}}{\partial y_{3}}}~{\cfrac {\partial y_{3}}{\partial \alpha }}\right]_{\alpha =0}=\left[{\cfrac {\partial {\boldsymbol {T}}}{\partial y_{1}}}~c_{1}+{\cfrac {\partial {\boldsymbol {T}}}{\partial y_{2}}}~c_{2}+{\cfrac {\partial {\boldsymbol {T}}}{\partial y_{3}}}~c_{3}\right]_{\alpha =0}\\&={\cfrac {\partial {\boldsymbol {T}}}{\partial x_{1}}}~c_{1}+{\cfrac {\partial {\boldsymbol {T}}}{\partial x_{2}}}~c_{2}+{\cfrac {\partial {\boldsymbol {T}}}{\partial x_{3}}}~c_{3}\equiv {\cfrac {\partial {\boldsymbol {T}}}{\partial x_{i}}}~c_{i}={\cfrac {\partial {\boldsymbol {T}}}{\partial x_{i}}}~(\mathbf {e} _{i}\cdot \mathbf {c} )=\left[{\cfrac {\partial {\boldsymbol {T}}}{\partial x_{i}}}\otimes \mathbf {e} _{i}\right]\cdot \mathbf {c} \qquad \square \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b848452c7189fb099eb1dadf20cd7c167dd4d2e)
![{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\phi ={}\quad &{\frac {\partial \phi }{\partial r}}~\mathbf {e} _{r}+{\frac {1}{r}}~{\frac {\partial \phi }{\partial \theta }}~\mathbf {e} _{\theta }+{\frac {\partial \phi }{\partial z}}~\mathbf {e} _{z}\\{\boldsymbol {\nabla }}\mathbf {v} ={}\quad &{\frac {\partial v_{r}}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\frac {1}{r}}\left({\frac {\partial v_{r}}{\partial \theta }}-v_{\theta }\right)~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }+{\frac {\partial v_{r}}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\\{}+{}&{\frac {\partial v_{\theta }}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\frac {1}{r}}\left({\frac {\partial v_{\theta }}{\partial \theta }}+v_{r}\right)~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }+{\frac {\partial v_{\theta }}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\\{}+{}&{\frac {\partial v_{z}}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\frac {1}{r}}{\frac {\partial v_{z}}{\partial \theta }}~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }+{\frac {\partial v_{z}}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\\{\boldsymbol {\nabla }}{\boldsymbol {S}}={}\quad &{\frac {\partial S_{rr}}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\frac {\partial S_{rr}}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{z}+{\frac {1}{r}}\left[{\frac {\partial S_{rr}}{\partial \theta }}-(S_{\theta r}+S_{r\theta })\right]~\mathbf {e} _{r}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\\{}+{}&{\frac {\partial S_{r\theta }}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\frac {\partial S_{r\theta }}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{z}+{\frac {1}{r}}\left[{\frac {\partial S_{r\theta }}{\partial \theta }}+(S_{rr}-S_{\theta \theta })\right]~\mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\\{}+{}&{\frac {\partial S_{rz}}{\partial r}}~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\frac {\partial S_{rz}}{\partial z}}~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{z}+{\frac {1}{r}}\left[{\frac {\partial S_{rz}}{\partial \theta }}-S_{\theta z}\right]~\mathbf {e} _{r}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\\{}+{}&{\frac {\partial S_{\theta r}}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\frac {\partial S_{\theta r}}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{z}+{\frac {1}{r}}\left[{\frac {\partial S_{\theta r}}{\partial \theta }}+(S_{rr}-S_{\theta \theta })\right]~\mathbf {e} _{\theta }\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\\{}+{}&{\frac {\partial S_{\theta \theta }}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\frac {\partial S_{\theta \theta }}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{z}+{\frac {1}{r}}\left[{\frac {\partial S_{\theta \theta }}{\partial \theta }}+(S_{r\theta }+S_{\theta r})\right]~\mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\\{}+{}&{\frac {\partial S_{\theta z}}{\partial r}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\frac {\partial S_{\theta z}}{\partial z}}~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{z}+{\frac {1}{r}}\left[{\frac {\partial S_{\theta z}}{\partial \theta }}+S_{rz}\right]~\mathbf {e} _{\theta }\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\\{}+{}&{\frac {\partial S_{zr}}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{r}+{\frac {\partial S_{zr}}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{z}+{\frac {1}{r}}\left[{\frac {\partial S_{zr}}{\partial \theta }}-S_{z\theta }\right]~\mathbf {e} _{z}\otimes \mathbf {e} _{r}\otimes \mathbf {e} _{\theta }\\{}+{}&{\frac {\partial S_{z\theta }}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{r}+{\frac {\partial S_{z\theta }}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{z}+{\frac {1}{r}}\left[{\frac {\partial S_{z\theta }}{\partial \theta }}+S_{zr}\right]~\mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\otimes \mathbf {e} _{\theta }\\{}+{}&{\frac {\partial S_{zz}}{\partial r}}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{r}+{\frac {\partial S_{zz}}{\partial z}}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{z}+{\frac {1}{r}}~{\frac {\partial S_{zz}}{\partial \theta }}~\mathbf {e} _{z}\otimes \mathbf {e} _{z}\otimes \mathbf {e} _{\theta }\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f19df7af85562767daf41dcce9042d343a7cf917)
![{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\cdot \left({\boldsymbol {\nabla }}\mathbf {v} \right)&={\boldsymbol {\nabla }}\cdot \left(v_{i,j}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\right)=v_{i,ji}~\mathbf {e} _{i}\cdot \mathbf {e} _{i}\otimes \mathbf {e} _{j}=\left({\boldsymbol {\nabla }}\cdot \mathbf {v} \right)_{,j}~\mathbf {e} _{j}={\boldsymbol {\nabla }}\left({\boldsymbol {\nabla }}\cdot \mathbf {v} \right)\\{\boldsymbol {\nabla }}\cdot \left[\left({\boldsymbol {\nabla }}\mathbf {v} \right)^{\textsf {T}}\right]&={\boldsymbol {\nabla }}\cdot \left(v_{j,i}~\mathbf {e} _{i}\otimes \mathbf {e} _{j}\right)=v_{j,ii}~\mathbf {e} _{i}\cdot \mathbf {e} _{i}\otimes \mathbf {e} _{j}={\boldsymbol {\nabla }}^{2}v_{j}~\mathbf {e} _{j}={\boldsymbol {\nabla }}^{2}\mathbf {v} \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/864380cd0a82178354a80ee58109fc0519c149ba)
![{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}&=\left[{\cfrac {\partial S_{ij}}{\partial q^{k}}}-\Gamma _{ki}^{l}~S_{lj}-\Gamma _{kj}^{l}~S_{il}\right]~g^{ik}~\mathbf {b} ^{j}\\[8pt]&=\left[{\cfrac {\partial S^{ij}}{\partial q^{i}}}+\Gamma _{il}^{i}~S^{lj}+\Gamma _{il}^{j}~S^{il}\right]~\mathbf {b} _{j}\\[8pt]&=\left[{\cfrac {\partial S_{~j}^{i}}{\partial q^{i}}}+\Gamma _{il}^{i}~S_{~j}^{l}-\Gamma _{ij}^{l}~S_{~l}^{i}\right]~\mathbf {b} ^{j}\\[8pt]&=\left[{\cfrac {\partial S_{i}^{~j}}{\partial q^{k}}}-\Gamma _{ik}^{l}~S_{l}^{~j}+\Gamma _{kl}^{j}~S_{i}^{~l}\right]~g^{ik}~\mathbf {b} _{j}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85fad1b0228d189867b87f1d770c3968e4394860)
![{\displaystyle {\begin{aligned}{\boldsymbol {\nabla }}\cdot \mathbf {v} =\quad &{\frac {\partial v_{r}}{\partial r}}+{\frac {1}{r}}\left({\frac {\partial v_{\theta }}{\partial \theta }}+v_{r}\right)+{\frac {\partial v_{z}}{\partial z}}\\{\boldsymbol {\nabla }}\cdot {\boldsymbol {S}}=\quad &{\frac {\partial S_{rr}}{\partial r}}~\mathbf {e} _{r}+{\frac {\partial S_{r\theta }}{\partial r}}~\mathbf {e} _{\theta }+{\frac {\partial S_{rz}}{\partial r}}~\mathbf {e} _{z}\\{}+{}&{\frac {1}{r}}\left[{\frac {\partial S_{\theta r}}{\partial \theta }}+(S_{rr}-S_{\theta \theta })\right]~\mathbf {e} _{r}+{\frac {1}{r}}\left[{\frac {\partial S_{\theta \theta }}{\partial \theta }}+(S_{r\theta }+S_{\theta r})\right]~\mathbf {e} _{\theta }+{\frac {1}{r}}\left[{\frac {\partial S_{\theta z}}{\partial \theta }}+S_{rz}\right]~\mathbf {e} _{z}\\{}+{}&{\frac {\partial S_{zr}}{\partial z}}~\mathbf {e} _{r}+{\frac {\partial S_{z\theta }}{\partial z}}~\mathbf {e} _{\theta }+{\frac {\partial S_{zz}}{\partial z}}~\mathbf {e} _{z}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8cd23836a8e6cc12150592c3964d95d6a3f94e9)
![{\displaystyle {\frac {\partial }{\partial {\boldsymbol {A}}}}\det({\boldsymbol {A}})=\det({\boldsymbol {A}})~\left[{\boldsymbol {A}}^{-1}\right]^{\textsf {T}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a229cf1ec76d8d0d6c4ebf0e55e24a9289524d0f)
![{\displaystyle {\begin{aligned}{\frac {\partial f}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}&=\left.{\cfrac {d}{d\alpha }}\det({\boldsymbol {A}}+\alpha ~{\boldsymbol {T}})\right|_{\alpha =0}\\&=\left.{\cfrac {d}{d\alpha }}\det \left[\alpha ~{\boldsymbol {A}}\left({\cfrac {1}{\alpha }}~{\boldsymbol {\mathit {I}}}+{\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\right)\right]\right|_{\alpha =0}\\&=\left.{\cfrac {d}{d\alpha }}\left[\alpha ^{3}~\det({\boldsymbol {A}})~\det \left({\cfrac {1}{\alpha }}~{\boldsymbol {\mathit {I}}}+{\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\right)\right]\right|_{\alpha =0}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a280dfed26c159c4b698ac9db1c9e7b4ff9ce7c)
![{\displaystyle {\begin{aligned}{\frac {\partial f}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}&=\left.{\cfrac {d}{d\alpha }}\left[\alpha ^{3}~\det({\boldsymbol {A}})~\left({\cfrac {1}{\alpha ^{3}}}+I_{1}\left({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\right)~{\cfrac {1}{\alpha ^{2}}}+I_{2}\left({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\right)~{\cfrac {1}{\alpha }}+I_{3}\left({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\right)\right)\right]\right|_{\alpha =0}\\&=\left.\det({\boldsymbol {A}})~{\cfrac {d}{d\alpha }}\left[1+I_{1}\left({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\right)~\alpha +I_{2}\left({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\right)~\alpha ^{2}+I_{3}\left({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\right)~\alpha ^{3}\right]\right|_{\alpha =0}\\&=\left.\det({\boldsymbol {A}})~\left[I_{1}({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}})+2~I_{2}\left({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\right)~\alpha +3~I_{3}\left({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\right)~\alpha ^{2}\right]\right|_{\alpha =0}\\&=\det({\boldsymbol {A}})~I_{1}\left({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\right)~.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc69b4584b67f7559a984e1c3d5dfa5b0f663c83)
![{\displaystyle {\frac {\partial f}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}=\det({\boldsymbol {A}})~{\text{tr}}\left({\boldsymbol {A}}^{-1}\cdot {\boldsymbol {T}}\right)=\det({\boldsymbol {A}})~\left[{\boldsymbol {A}}^{-1}\right]^{\textsf {T}}:{\boldsymbol {T}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcfbbc8e843ee49828711668186ed4e5faa93384)
![{\displaystyle {\frac {\partial f}{\partial {\boldsymbol {A}}}}=\det({\boldsymbol {A}})~\left[{\boldsymbol {A}}^{-1}\right]^{\textsf {T}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf1328c61318cc48aa33f6c0b580b55e268f7ebe)
![{\displaystyle {\begin{aligned}I_{1}({\boldsymbol {A}})&={\text{tr}}{\boldsymbol {A}}\\I_{2}({\boldsymbol {A}})&={\frac {1}{2}}\left[({\text{tr}}{\boldsymbol {A}})^{2}-{\text{tr}}{{\boldsymbol {A}}^{2}}\right]\\I_{3}({\boldsymbol {A}})&=\det({\boldsymbol {A}})\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb5f440de0bb33a949001c6bef13f9f829fb1a42)
![{\displaystyle {\begin{aligned}{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}&={\boldsymbol {\mathit {1}}}\\[3pt]{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}&=I_{1}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}\\[3pt]{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}&=\det({\boldsymbol {A}})~\left[{\boldsymbol {A}}^{-1}\right]^{\textsf {T}}=I_{2}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}~\left(I_{1}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {A}}^{\textsf {T}}\right)=\left({\boldsymbol {A}}^{2}-I_{1}~{\boldsymbol {A}}+I_{2}~{\boldsymbol {\mathit {1}}}\right)^{\textsf {T}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19cf1ad5bce9774bf510c8818f4b90e32c4f2640)
![{\displaystyle {\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}=(A)~[A^{-}1]^{T}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ec1a4623eec8ccfc6052322375065a0a3718b7a)
![{\displaystyle {\frac {\partial }{\partial {\boldsymbol {A}}}}\det(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})=\det(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})~\left[(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})^{-1}\right]^{\textsf {T}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4be1b29969a25190e4efad41db9d76f11e8a8079)
![{\displaystyle {\begin{aligned}{\frac {\partial }{\partial {\boldsymbol {A}}}}\det(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})&={\frac {\partial }{\partial {\boldsymbol {A}}}}\left[\lambda ^{3}+I_{1}({\boldsymbol {A}})~\lambda ^{2}+I_{2}({\boldsymbol {A}})~\lambda +I_{3}({\boldsymbol {A}})\right]\\&={\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda +{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}~.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a4ad226ea72ae236eaddfe419007ff6de53d55d)
![{\displaystyle {\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda +{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}=\det(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})~\left[(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})^{-1}\right]^{\textsf {T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3993a79ec1f95da1b9e188243300861ee7ee2e45)
![{\displaystyle (\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})^{\textsf {T}}\cdot \left[{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda +{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}\right]=\det(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}})~{\boldsymbol {\mathit {1}}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/782fb846f7870890e72cda9dbeba6e80dfc064b0)
![{\displaystyle \left(\lambda ~{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}}^{\textsf {T}}\right)\cdot \left[{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda +{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}\right]=\left[\lambda ^{3}+I_{1}~\lambda ^{2}+I_{2}~\lambda +I_{3}\right]{\boldsymbol {\mathit {1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d9d08c98dc866bcc330678342ca3c391da0042a)
![{\displaystyle {\begin{aligned}\left[{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{3}\right.&\left.+{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}~\lambda \right]{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda +{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}\\&=\left[\lambda ^{3}+I_{1}~\lambda ^{2}+I_{2}~\lambda +I_{3}\right]{\boldsymbol {\mathit {1}}}~.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6174aae82fa111cbe2235a21c275bb7bc0e243b4)
![{\displaystyle {\begin{aligned}\left[{\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{3}\right.&\left.+{\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}~\lambda +{\frac {\partial I_{4}}{\partial {\boldsymbol {A}}}}\right]{\boldsymbol {\mathit {1}}}+{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{0}}{\partial {\boldsymbol {A}}}}~\lambda ^{3}+{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{1}}{\partial {\boldsymbol {A}}}}~\lambda ^{2}+{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{2}}{\partial {\boldsymbol {A}}}}~\lambda +{\boldsymbol {A}}^{\textsf {T}}\cdot {\frac {\partial I_{3}}{\partial {\boldsymbol {A}}}}\\&=\left[I_{0}~\lambda ^{3}+I_{1}~\lambda ^{2}+I_{2}~\lambda +I_{3}\right]{\boldsymbol {\mathit {1}}}~.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8169e3ea8470a718a349e0d8a73e1fb7c162914)
![{\displaystyle {\frac {\partial {\boldsymbol {A}}}{\partial {\boldsymbol {A}}}}:{\boldsymbol {T}}=\left[{\frac {\partial }{\partial \alpha }}({\boldsymbol {A}}+\alpha ~{\boldsymbol {T}})\right]_{\alpha =0}={\boldsymbol {T}}={\boldsymbol {\mathsf {I}}}:{\boldsymbol {T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4cf9341eabbe69c48f4ff85db571b84c8b2c318)
![{\displaystyle {\frac {\partial }{\partial {\boldsymbol {S}}}}[{\boldsymbol {F}}_{1}({\boldsymbol {S}})\cdot {\boldsymbol {F}}_{2}({\boldsymbol {S}})]:{\boldsymbol {T}}=\left({\frac {\partial {\boldsymbol {F}}_{1}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)\cdot {\boldsymbol {F}}_{2}+{\boldsymbol {F}}_{1}\cdot \left({\frac {\partial {\boldsymbol {F}}_{2}}{\partial {\boldsymbol {S}}}}:{\boldsymbol {T}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73a25e5e0ee3f8a2f287da104d5f72d8342899b9)
Datos: Q7700706