Undergrad Calculating Jacobian Determinant

[Apashanka]

I came across a line in this paper at page (2) at right side 2nd para where it is written $d^3x=Jd^3X$ where $J$ is the Jacobian and x and X are the positions of the fluid elements at time $t_0$ and $t$ respectively.
Here what I have concluded that $x_i=f(X_i)$ where the functional dependence of $X_i$(e.g $f(X_i)$) varies with the time evolution of $x$ for the $i^{th}$ coordinate and call this $f_i$ and now using this $dx_i=\frac{\partial f_i}{\partial X_i}dX_i$ and similarly for the $i^{th},j^{th}$ and $k^{th}$.
The matrix containing the terms $\frac{\partial f_i}{\partial x_i}$ at the diagonal is the Jacobian matrix and it's determinant times $d^3X$ gives $d^3x$
Isn't it??

 

[vanhees71]

It's the Jacobian. Your notation is very dangerous. There must never ever be equations with more than two equal indices. If there are two equal indices you have to sum over them, i.e.,
$$\mathrm{d} x_i = \frac{\partial x_i}{\partial X_j} \mathrm{d} X_j=\sum_{j=1}^3\frac{\partial x_i}{\partial X_j} \mathrm{d} X_j ,$$
and the Jacobian is the full determinand of the Jacobian Matrix,
$$J=\mathrm{det} \frac{\partial x_i}{\partial X_j}.$$
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