Let
be a surface. Let
be the position vector of points on the surface and let
be a vector field that are defined on
. If
![{\displaystyle \int _{\partial {\Omega }}\mathbf {x} \times \mathbf {t} ~{\text{dA}}=\mathbf {0} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/5dff9a7d3dab5642185b38568c1f39bf0ae5a241)
show that
![{\displaystyle \int _{\partial {\Omega }}\mathbf {x} \otimes \mathbf {t} ~{\text{dA}}=\int _{\partial {\Omega }}\mathbf {t} \otimes \mathbf {x} ~{\text{dA}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01955704832d99e83d3be8f682469de46ff5321a)
If we assume a Cartesian basis, we can write the given relation in index
notation as
![{\displaystyle \int _{\partial {\Omega }}e_{ijk}~x_{j}~t_{k}~{\text{dA}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce66bab89a8533be73ca423b4e27da1a9d72c574)
where
is the Levi-Civita (permutation) symbol.
Since
is does not depend upon the position we can write
![{\displaystyle e_{ijk}\int _{\partial {\Omega }}~x_{j}~t_{k}~{\text{dA}}=0~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4f8d3ba1dee3f617edc2ec937f7c71c9c264551)
Define
![{\displaystyle A_{jk}:=\int _{\partial {\Omega }}x_{j}~t_{k}~{\text{dA}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/194e401300aa2a7fa8345d7afa6ffdc741c3ced9)
Then,
![{\displaystyle e_{ijk}A_{jk}=0~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/168fe9108f55fa3f9f091b70dac9b084bb4dcb5d)
Expanding, we get
![{\displaystyle e_{i11}~~A_{11}+e_{i12}A_{12}+e_{i13}A_{13}+e_{i21}A_{21}+e_{i22}~~A_{22}+e_{i23}A_{23}+e_{i31}A_{31}+e_{i32}A_{32}+e_{i33}~~A_{33}=0\qquad i=1,2,3~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e241c3879f46d3a0a9a0e6f2d1f1281f1f4ab491)
Expanding further, we get three equations
![{\displaystyle {\begin{aligned}e_{112}A_{12}+e_{113}A_{13}+e_{121}A_{21}+e_{123}A_{23}+e_{131}A_{31}+e_{132}A_{32}&=0\\e_{212}A_{12}+e_{213}A_{13}+e_{221}A_{21}+e_{223}A_{23}+e_{231}A_{31}+e_{232}A_{32}&=0\\e_{312}A_{12}+e_{313}A_{13}+e_{321}A_{21}+e_{323}A_{23}+e_{331}A_{31}+e_{332}A_{32}&=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf9166c7aea772437a783b823e40f8c7bf5e78c8)
or,
![{\displaystyle {\begin{aligned}e_{123}A_{23}+e_{132}A_{32}&=0\\e_{213}A_{13}+e_{231}A_{31}&=0\\e_{312}A_{12}+e_{321}A_{21}&=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7197bb700823495884ebb3c6c4241eeafbbaa4ff)
or,
![{\displaystyle A_{23}=A_{32}~;~~A_{13}=A_{31}~;~~A_{12}=A_{21}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e466bd902d12dd7c589b97d84dfffd2bfdcbcf20)
Therefore,
![{\displaystyle \int _{\partial {\Omega }}x_{2}~t_{3}~{\text{dA}}=\int _{\partial {\Omega }}x_{3}~t_{2}~{\text{dA}}=\int _{\partial {\Omega }}t_{2}~x_{3}~{\text{dA}}~;~~\int _{\partial {\Omega }}x_{1}~t_{3}~{\text{dA}}=\int _{\partial {\Omega }}x_{3}~t_{1}~{\text{dA}}=\int _{\partial {\Omega }}t_{1}~x_{3}~{\text{dA}}~;~~\int _{\partial {\Omega }}x_{1}~t_{2}~{\text{dA}}=\int _{\partial {\Omega }}x_{2}~t_{1}~{\text{dA}}=\int _{\partial {\Omega }}t_{1}~x_{2}~{\text{dA}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88f9fb35b50e176c04049f70cb66fbdff35af041)
Also, by symmetry,
![{\displaystyle \int _{\partial {\Omega }}x_{1}~t_{1}~{\text{dA}}=\int _{\partial {\Omega }}t_{1}~x_{1}~{\text{dA}}~;~~\int _{\partial {\Omega }}x_{2}~t_{2}~{\text{dA}}=\int _{\partial {\Omega }}t_{2}~x_{2}~{\text{dA}}~;~~\int _{\partial {\Omega }}x_{3}~t_{3}~{\text{dA}}=\int _{\partial {\Omega }}t_{3}~x_{3}~{\text{dA}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f19a5dfa1654a3e628ea49bbe4baf05d944a03f7)
Therefore we may write,
![{\displaystyle \int _{\partial {\Omega }}x_{j}~t_{k}~{\text{dA}}=\int _{\partial {\Omega }}t_{j}~x_{k}~{\text{dA}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f50424992df963108451381013c5ba42f1e24cc0)
Reverting back to direct tensor notation, we get
![{\displaystyle {\int _{\partial {\Omega }}\mathbf {x} \otimes \mathbf {t} ~{\text{dA}}=\int _{\partial {\Omega }}\mathbf {t} \otimes \mathbf {x} ~{\text{dA}}~.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5145efacd47887732562b708d998fcc93e6c619)