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Metric Discriminant

Given a metric g_(alphabeta), the discriminant is defined by

g=det(g_(alphabeta))
(1)
=|g_(11) g_(12); g_(21) g_(22)|
(2)
=g_(11)g_(22)-(g_(12))^2.
(3)

Let g be the discriminant and g^_ the transformed discriminant, then

g^_=D^2g
(4)
g=D^_^2g^_,
(5)

where

D=(partial(u^1,u^2))/(partial(u^_^1,u^_^2))=|(partialu^1)/(partialu^_^1) (partialu^1)/(partialu^_^2); (partialu^2)/(partialu^_^1) (partialu^2)/(partialu^_^2)|
(6)
D^_=(partial(u^_^1,u^_^2))/(partial(u^1,u^2))=|(partialu^_^1)/(partialu^1) (partialu^_^1)/(partialu^2); (partialu^_^2)/(partialu^1) (partialu^_^2)/(partialu^2)|.
(7)

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Cite this as:

Weisstein, Eric W. "Metric Discriminant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MetricDiscriminant.html

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