Talk:Nonlinear finite elements/Lagrangian finite elements

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1. To clarify Jacobian determinant

Jacobian determinant is the same as the determinant of the deformation gradient

It's often simply called the Jacobian as well (see here a good explanation why is called Jacobian, because

"This is such an important result that is given a special symbol, , and a special name, the Jacobian."

2. It's also a bit difficult to understand how this equation is derived. Since it we start from the definition of Jacobian, it will be

which is different from the presented equation .

But I found an explanation from Ted Belytschko’s book, p.22, explaining how is derived.

The Jocobian is usually defined by for one-dimensional maps. However, to maintain the consistency with multi-dimensional formulations, we will define the Jacobian as the ratio of an infinitesimal volume in the deformed body, , to the corresponding volume of the segment in the undeformed body :

If we substitute into above equation, we will get ( as it is a scalar for this 1D problem)

which is consistent to the interpretation of as volume ratio.