What is the dual of a tensor and where does the -1/6 come from?

[yukcream]

In my lecture note about tensor it states:
As element of integration over the hypersurface it is more convenient to use the 4-vector dS^i , dual to the tensor dS^ikl:

dS^i = -1/6 e^iklmdS_klm ,dS_klm = e_iklmdS^i

What is the meaning of dual?
How the -1/6 comes from?

yukyuk

 

[robphy]

You have contracted over three indices involving totally antisymmetric tensors. 3!=6. The (-1) comes from the metric signature. See https://www.physicsforums.com/showthread.php?t=90239 for a related discussion.