# Talk:Nonlinear finite elements/Lagrangian finite elements

1. To clarify Jacobian determinant

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

${\displaystyle J=\det {\boldsymbol {F}}}$

It's often simply called the Jacobian as well (see here a good explanation why ${\displaystyle \det {\boldsymbol {F}}}$ is called Jacobian, because

"This is such an important result that ${\displaystyle \det {\boldsymbol {F}}}$ is given a special symbol, ${\displaystyle J}$, and a special name, the Jacobian."

2. It's also a bit difficult to understand how this equation ${\displaystyle {J={\cfrac {A}{A_{0}}}F}}$ is derived. Since it we start from the definition of Jacobian, it will be

${\displaystyle J=\det {\boldsymbol {F}}=\det {\frac {\partial x}{\partial X}}=\det {\frac {\partial ({\frac {L}{L_{0}}}X)}{\partial X}}=\det {\frac {L}{L_{0}}}={\frac {L}{L_{0}}}}$

which is different from the presented equation ${\displaystyle {J={\cfrac {A}{A_{0}}}F}}$.

But I found an explanation from Ted Belytschko’s book, p.22, explaining how ${\displaystyle {J={\cfrac {A}{A_{0}}}F}}$ is derived.

The Jocobian is usually defined by ${\displaystyle J=\det {\boldsymbol {F}}={\frac {\partial x}{\partial X}}~}$ 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, ${\displaystyle A\Delta x}$, to the corresponding volume of the segment in the undeformed body ${\displaystyle A_{0}\Delta X}$:

${\displaystyle {J={\frac {\partial x}{\partial X}}{\frac {A}{A_{0}}}={\cfrac {A}{A_{0}}}F~}}$

If we substitute ${\displaystyle \det {\boldsymbol {F}}={\frac {L}{L_{0}}}}$ into above equation, we will get (${\displaystyle \det {\boldsymbol {F}}=F~}$ as it is a scalar for this 1D problem)

${\displaystyle {J={\cfrac {A}{A_{0}}}F={\cfrac {A}{A_{0}}}{\frac {L}{L_{0}}}={\frac {V}{V_{0}}}}}$

which is consistent to the interpretation of ${\displaystyle J}$ as volume ratio.