- #1
Apashanka
- 429
- 15
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??
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??