Jacobian, Complex Analysis | Basic Physics for IIT JAM PDF Download

Although we will not prove it, the general result for a change of coordinates in an n-dimensional integral from a set x_i to a set y_j (where i and j both run from 1 to n) is
where then-dimensional Jacobian can be written as an n×n determinant in an analogous way to the two- and three-dimensional cases.
For readers who already have sufficient familiarity with matrices and their properties, a fairly compact proof of some useful general properties of Jacobians can be given as follows. Other readers should turn straight to the results (6.16) and (6.17) and return to the proof at some later time.
Consider three sets of variables x_i,y_i and z_i, with i running from 1 to n for each set. From the chain rule in partial differentiation , we know that (6.13) 

Now let A,B and C be the matrices whose ij^th elements are∂xi/∂y_j,∂y_i/∂z_j and∂x_i/∂z_j respectively. We can then write (6.13) as the matrix product  (6.14) 

We may now use the general result for the determinant of the product of two matrices, namely |AB|=|A||B|, and recall that the Jacobian  (6.15) 

and similarly for J_yz and J_xz. On taking the determinant of (6.14), we therefore obtain 

J_xz=J_xy J_yz 

or, in the usual notation,

 (6.16) 

As a special case, if the set z_i is taken to be identical to the set x_i,and the obvious result J_xx= 1 is used, we obtain 

J_xy J_yx=1 

or, in the usual notation, 

 (6.17) 

The similarity between the properties of Jacobians and those of derivatives is apparent, and to some extent is suggested by the notation. We further note from(6.15) that since|A|=|A^T|,where A^T is the transpose of A, we can interchange the rows and columns in the determinantal form of the Jacobian without changing its value.