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So I'm learning tensors and found a thing i can't quite understand:
The formula for 3d cross product is as follows:
[eqn]c^i=\epsilon _{ijk}a^jb^k[/eqn]
However, applying tensor transformation rule yields
[eqn]\frac{\partial{x^i'}}{\partial{x^i}}}c^i=\frac{\partial{x^i}}{\partial{x^i'}}}\epsilon _{ijk}a^jb^k[/eqn]
, because LCT is of rank (0,3).
Where did I fuck up?
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basically [eqn]{\partial{x^i'}}{\partial{x^i}}={\partial{x^i}}{\partial{x^i'}}[eqn/]
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[eqn]{\partial{x^i'}}{\partial{x^i}}={\partial{x^i}}{\partial{x^i'}}[/eqn]
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the first equation is ill formed since c should be covariant
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>>8930864
also the levi-civita is a rank 3 object, but it is NOT a tensor. It's called a symbol similar to the Christoffel symbols
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>>8930864
Why? Isn't it a vector?
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>>8930868
from what i've seen, it's called a symbol when used in a single coordinate system and a tensor when we want to transform an expression into another coordintae system like in the OP
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>>8930850
You're using the wrong right-hand rule.
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>>8930880
Ok thanks, this is very confusing to call it a tensor then.
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levi civita symbol is a tensor density of weight one, however you can naturally make any tensor density into a tensor by scaling it my a metric determinant
[eqn]c_{i} = \sqrt{g} \varepsilon_{ijk}a^{j}b^{k}[/eqn]
this will transform correctly and is the 'better' definition of the cross product.
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>>8930891
don't think people do lol
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>>8930894
this is pretty neat
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>>8930894
but it requires an additional structure
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>>8931288
the metric is the key to covariance, which is the point of tensor analysis.
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