Metric Discriminant

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

g = det(g_(alphabeta))

(1)

= [画像:|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 = [画像:(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^_ = [画像:(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)

Explore with Wolfram|Alpha

WolframAlpha

More things to try:

Cite this as:

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

Subject classifications