Jacobian reminder

The Jacobian is initially an expression that helps to compute a change in coordinate space for integration.

From carthesian to polar we get dxdy = r*drdO as illustrated in the following figure. The jacobian is considered to be equal to r.

More generally, if dxdy = d(u,v)/d(x,y)*dudv,  d(u,v)/d(x,y) is the Jacobian.
In two dimention, the general formula for the Jacobian is

J = dx/du * dy/dv  -  dy/du * dy/dv

It corresponds to the determinant of the matrix of partial derivation (this matrix is also called the Jacobian sometimes).
 

 | dx/du dy/du |  
J = det   |           |      =  dx/du * dy/dv  -  dy/du * dy/dv 

 | dx/dv dy/dv |

It can be generalized to any dimention.
 

 | dx/du dy/du     dz/du |  
J = det   | dx/dv dy/dv     dz/dv |  =  dx/du *( dy/dv*dx/dw -  dy/dw*dz/dv )  + dy/du *( dz/dv*dx/dw

 | dx/dw dy/dw    dzdw |               -  dzdw*dx/dv ) + dzdu *( dx/dv*dy/dw -  dx/dw*dy/dv ) 

From a carthesian 3D to spherical, one get J = (r^2)*sin(O)

Physical meaning of the Jacobian

The determinant represents the volume depicted by the rows of the matrix views as vectors (see determinant). Volume here is taken in the general sens and does not depend on the dimentionnality (in 1D, volume=length, in 2D, volume=area, in 3D, volume=actual 3D volume, in 4D, volume =  ...).
Thus, here, the jacobian represents the transformation of one volume unit from one coordinate space to another. For instance, in 2D carthesian coordinate, a volume (surface) of 1 correspond to a volume of r in polar coordinates. In 3D carthesian coordinate space, a volume of 1 correspond to a volume of (r^2)*sin(O) in spherical coordinate space (for this transformation J = (r^2)*sin(O)).

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