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December 1st, 2014, 07:49 AM   #1
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Proof that V/V0 = det(F)

I wish to prove in index notation that the change in volume expressed as a ratio is equal to the determinant of the deformation tensor, F.

x1' = F x1
x2' = F x2
x3' = F x3

Original volume =

V0 = x1 . (x2 x x3) = Ɛijk x1i x2j x3k

Deformed volume =

V = x1' . (x2' x x3') = Ɛlmn x1i x2j x3k Fli Fmj Fnk

Which is as far as i've gotten so how does V/V0 = det(F) i.e.,

Ɛijk Fi1 Fj2 Fk3

Many thanks
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