Continuity equation in the Lagrangian flow picture approach

Question

In deriving continuity equations using Lagrangian.

We consider the element of fluid which occupied a rectangular parallelopiped having its centre at the point $(a,b,c)$ and its edges $delta a$ , $delta b$ ,$delta c $ parallel to the axes . At the time $t$ the same element for an oblique parallelepiped . The centre now has for its co-ordinates $x$ , $y$ , $z  ;$ and the projections of the edges on the co-ordinate axes are respectively 
$$ frac{partial x}{partial a} delta a  ,  frac{partial y}{partial a} delta a  ,  frac{partial z}{partial c} delta a$$
$$frac{partial x}{partial b} delta b  ,  frac{partial y}{partial b} delta b  ,  frac{partial z}{partial b} delta b$$
$$frac{partial x}{partial c} delta c  ,  frac{partial y}{partial c} delta c  ,  frac{partial z}{partial c} delta c$$

How can i get these projections ?
The volume of the parallelepiped is therefore 
$$begin{vmatrix}
frac{partial x}{partial a} & frac{partial y}{partial a} & frac{partial z}{partial a}  
frac{partial x}{partial b} & frac{partial y}{partial b} & frac{partial z}{partial b}  
frac{partial x}{partial c} & frac{partial y}{partial c} & frac{partial z}{partial c}
end{vmatrix} delta a delta b delta c$$
or as its often written $$frac{D(x,y,z)}{D(a,b,c)} delta a delta b delta c$$

since the fluid mass is unchanged and the fluid is incompressble we have 
$$frac{D(x,y,z)}{D(a,b,c)} =1$$

Is there a way to prove that $$frac{D(a,b,c)}{D(x,y,z)}= 1$$
without expanding the determinant?

Qmechanic · Answer

In the Lagrangian flow picture ${bf a}equiv(a,b,c)$ typically denote continuous labels of a fluid parcel distributed such that
$$d(text{mass})~=~da~db~dc,tag{2.1}$$
cf. e.g. Ref. 1.

On the other hand ${bf x}equiv(x,y,z)$ typically denote the position coordinates of a fluid parcel. Therefore the mass density becomes
$$ rho~=~ |detfrac{partial{bf a}}{partial{bf x}} |.tag{2.2}$$

The flow velocity is defined as $${bf u}~equiv~ frac{d{bf x}}{dt}.tag{2.4}$$ The mass continuum equation follows in the Lagrangian flow picture from
$$begin{align} -frac{dlnrho}{dt}~=~&frac{d}{dt}ln |det frac{partial{bf x}}{partial{bf a}}|cr
~=~&{rm tr}left( frac{partial{bf a}}{partial{bf x}}frac{d}{dt}frac{partial{bf x}}{partial{bf a}}right)cr
~=~&{rm tr}left( frac{partial}{partial{bf x}}frac{d{bf x}}{dt}right)cr
~=~&{bf nabla}cdot {bf u}.end{align}  tag{2.3}$$
For an incompressible flow, the density $rho$ is constant along the fluid flow.

References:

R. Salmon, Hamiltonian Fluid Mechanics, Ann. Rev Fluid. Mech. (1988) 225. The pdf file can be downloaded from the author's webpage.

Continuity equation in the Lagrangian flow picture approach

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