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En càlcul vectorial , el lema de Chandrasekhar-Wentzel va ser derivat per Subrahmanyan Chandrasekhar i Gregor Wentzel el 1965 mentre estudiaven l'estabilitat de la rotació d'una gota d'un líquid.[1] [2] si
S
{\displaystyle \mathbf {S} }
C
{\displaystyle C}
L
=
∮
C
x
×
(
d
x
×
n
)
=
−
∫
S
(
x
×
n
)
∇
⋅
n
d
S
.
{\displaystyle \mathbf {L} =\oint _{C}\mathbf {x} \times (d\mathbf {x} \times \mathbf {n} )=-\int _{\mathbf {S} }(\mathbf {x} \times \mathbf {n} )\nabla \cdot \mathbf {n} \ dS.}
Aquí,
x
{\displaystyle \mathbf {x} }
n
{\displaystyle \mathbf {n} }
S
{\displaystyle \mathbf {S} }
integral de línia tendeix a zero, donant lloc al resultat,
∫
S
(
x
×
n
)
∇
⋅
n
d
S
=
0
,
{\displaystyle \int _{\mathbf {S} }(\mathbf {x} \times \mathbf {n} )\nabla \cdot \mathbf {n} \ dS=0,}
o, en notació d'índex , tenim
∫
S
x
j
∇
⋅
n
d
S
k
=
∫
S
x
k
∇
⋅
n
d
S
j
.
{\displaystyle \int _{\mathbf {S} }x_{j}\nabla \cdot \mathbf {n} \ dS_{k}=\int _{\mathbf {S} }x_{k}\nabla \cdot \mathbf {n} \ dS_{j}.}
És a dir el tensor
T
i
j
=
∫
S
x
j
∇
⋅
n
d
S
i
{\displaystyle T_{ij}=\int _{\mathbf {S} }x_{j}\nabla \cdot \mathbf {n} \ dS_{i}}
definida en una superfície tancada és sempre simètrica, és a dir,
T
i
j
=
T
j
i
{\displaystyle T_{ij}=T_{ji}}
Escrivim el vector en notació d'índex , però s'evitarà la convenció de sumació a tota la demostració. Aleshores es pot escriure el costat esquerre
L
i
=
∮
C
[
d
x
i
(
n
i
x
j
+
n
k
x
k
)
+
d
x
j
(
−
n
i
x
j
)
+
d
x
k
(
−
n
i
x
k
)
]
.
{\displaystyle L_{i}=\oint _{C}[dx_{i}(n_{i}x_{j}+n_{k}x_{k})+dx_{j}(-n_{i}x_{j})+dx_{k}(-n_{i}x_{k})].}
Convertint la integral de línia en integral de superfície mitjançant el teorema de Stokes , obtenim
L
i
=
∫
S
{
n
i
[
∂
∂
x
j
(
−
n
i
x
k
)
−
∂
∂
x
k
(
−
n
i
x
j
)
]
+
n
j
[
∂
∂
x
k
(
n
j
x
j
+
n
k
x
k
)
−
∂
∂
x
i
(
−
n
i
x
k
)
]
+
n
k
[
∂
∂
x
i
(
−
n
i
x
j
)
−
∂
∂
x
j
(
n
j
x
j
+
n
k
x
k
)
]
}
d
S
.
{\displaystyle L_{i}=\int _{\mathbf {S} }\left\{n_{i}\left[{\frac {\partial }{\partial x_{j}}}(-n_{i}x_{k})-{\frac {\partial }{\partial x_{k}}}(-n_{i}x_{j})\right]+n_{j}\left[{\frac {\partial }{\partial x_{k}}}(n_{j}x_{j}+n_{k}x_{k})-{\frac {\partial }{\partial x_{i}}}(-n_{i}x_{k})\right]+n_{k}\left[{\frac {\partial }{\partial x_{i}}}(-n_{i}x_{j})-{\frac {\partial }{\partial x_{j}}}(n_{j}x_{j}+n_{k}x_{k})\right]\right\}\ dS.}
Realitzem la diferenciació necessària i després d'alguna reordenació, obtenim
L
i
=
∫
S
[
−
1
2
x
k
∂
∂
x
j
(
n
i
2
+
n
k
2
)
+
1
2
x
j
∂
∂
x
k
(
n
i
2
+
n
j
2
)
+
n
j
x
k
(
∂
n
i
∂
x
i
+
∂
n
k
∂
x
k
)
−
n
k
x
j
(
∂
n
i
∂
x
i
+
∂
n
j
∂
x
j
)
]
d
S
,
{\displaystyle L_{i}=\int _{\mathbf {S} }\left[-{\frac {1}{2}}x_{k}{\frac {\partial }{\partial x_{j}}}(n_{i}^{2}+n_{k}^{2})+{\frac {1}{2}}x_{j}{\frac {\partial }{\partial x_{k}}}(n_{i}^{2}+n_{j}^{2})+n_{j}x_{k}\left({\frac {\partial n_{i}}{\partial x_{i}}}+{\frac {\partial n_{k}}{\partial x_{k}}}\right)-n_{k}x_{j}\left({\frac {\partial n_{i}}{\partial x_{i}}}+{\frac {\partial n_{j}}{\partial x_{j}}}\right)\right]\ dS,}
o, dit d'una altra manera,
L
i
=
∫
S
[
1
2
(
x
j
∂
∂
x
k
−
x
k
∂
∂
x
j
)
|
n
|
2
−
(
x
j
n
k
−
x
k
n
j
)
∇
⋅
n
]
d
S
.
{\displaystyle L_{i}=\int _{\mathbf {S} }\left[{\frac {1}{2}}\left(x_{j}{\frac {\partial }{\partial x_{k}}}-x_{k}{\frac {\partial }{\partial x_{j}}}\right)|\mathbf {n} |^{2}-(x_{j}n_{k}-x_{k}n_{j})\nabla \cdot \mathbf {n} \right]\ dS.}
I llavors
|
n
|
2
=
1
{\displaystyle |\mathbf {n} |^{2}=1}
L
i
=
−
∫
S
(
x
j
n
k
−
x
k
n
j
)
∇
⋅
n
d
S
,
{\displaystyle L_{i}=-\int _{\mathbf {S} }(x_{j}n_{k}-x_{k}n_{j})\nabla \cdot \mathbf {n} \ dS,}
demostrant així el lema.
↑ Chandrasekhar , SProceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences , 286(1404), 1965, pàg. 1–26. DOI : 10.1098/rspa.1965.0127 .↑ Chandrasekhar , S; Wali , K. C. A Quest for Perspectives: Selected Works of S. Chandrasekhar: With Commentary (en anglès). World Scientific, 2001.
![{\displaystyle L_{i}=\oint _{C}[dx_{i}(n_{i}x_{j}+n_{k}x_{k})+dx_{j}(-n_{i}x_{j})+dx_{k}(-n_{i}x_{k})].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09de3d8b9fcbd3a4e29791251a4d9e40e290e3e3)
![{\displaystyle L_{i}=\int _{\mathbf {S} }\left\{n_{i}\left[{\frac {\partial }{\partial x_{j}}}(-n_{i}x_{k})-{\frac {\partial }{\partial x_{k}}}(-n_{i}x_{j})\right]+n_{j}\left[{\frac {\partial }{\partial x_{k}}}(n_{j}x_{j}+n_{k}x_{k})-{\frac {\partial }{\partial x_{i}}}(-n_{i}x_{k})\right]+n_{k}\left[{\frac {\partial }{\partial x_{i}}}(-n_{i}x_{j})-{\frac {\partial }{\partial x_{j}}}(n_{j}x_{j}+n_{k}x_{k})\right]\right\}\ dS.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f21c635b5808af82e3f1f9b33415b223e496c488)
![{\displaystyle L_{i}=\int _{\mathbf {S} }\left[-{\frac {1}{2}}x_{k}{\frac {\partial }{\partial x_{j}}}(n_{i}^{2}+n_{k}^{2})+{\frac {1}{2}}x_{j}{\frac {\partial }{\partial x_{k}}}(n_{i}^{2}+n_{j}^{2})+n_{j}x_{k}\left({\frac {\partial n_{i}}{\partial x_{i}}}+{\frac {\partial n_{k}}{\partial x_{k}}}\right)-n_{k}x_{j}\left({\frac {\partial n_{i}}{\partial x_{i}}}+{\frac {\partial n_{j}}{\partial x_{j}}}\right)\right]\ dS,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f962a2a7c541656ccf0b64ca4b8149258a274364)
![{\displaystyle L_{i}=\int _{\mathbf {S} }\left[{\frac {1}{2}}\left(x_{j}{\frac {\partial }{\partial x_{k}}}-x_{k}{\frac {\partial }{\partial x_{j}}}\right)|\mathbf {n} |^{2}-(x_{j}n_{k}-x_{k}n_{j})\nabla \cdot \mathbf {n} \right]\ dS.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ece10fac16e521152b3c4f36eb1b07e906279d2)
