- #1
Telemachus
- 835
- 30
Homework Statement
I'm trying to demonstrate the dientity: ##\displaystyle \int_v \vec {r} \times ( \nabla \cdot T ) d^3x=\oint_S ( \vec r \times T) \cdot \hat n da##
T is a second order tensor, and r a vector.
So basically I should have: ##\vec {r} \times \nabla \cdot T= \nabla \cdot ( \vec r \times T)##
When I use indicial notation, I get a term: ##\epsilon_{ijk} T_{ki}## which should equal zero. I see in principle that if T is a symmetric tensor I get the desired result. But I think that the result should be general, and not hold only for a symmetrical tensor T.
What I did was:
##\epsilon_{ijk} T_{ki}=-\epsilon_{ikj} T_{ki}## (1)
and by the other side I've used that for dummy indices:
##\epsilon_{ijk} T_{ki}=\epsilon_{ikj} T_{ki}## (2)
This last step is the one that I'm not sure if its right. I thought that as the indices are dummies, I just could interchange in the Levi Civita symbol the j and the k, and then because of the properties of the levi civita symbol, use the indentity (1) to demonstrate that the term equals zero. But I wanted to know if (2) is fine, because I'm not really sure.
I think that actually if I use the dummy indices properly I should have: ##\epsilon_{ijk} T_{ki}=\epsilon_{ikj} T_{ji}## so what I did is actually wrong.
Thanks in advance.
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