Jacobian, Complex Analysis Physics Notes | EduRev

Basic Physics for IIT JAM

Physics : Jacobian, Complex Analysis Physics Notes | EduRev

The document Jacobian, Complex Analysis Physics Notes | EduRev is a part of the Physics Course Basic Physics for IIT JAM.
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Although we will not prove it, the general result for a change of coordinates in an n-dimensional integral from a set xi to a set yj (where and both run from 1 to n) isJacobian, Complex Analysis Physics Notes | EduRev
where then-dimensional Jacobian can be written as an 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 xi,yi and zi, with running from 1 to for each set. From the chain rule in partial differentiation , we know that Jacobian, Complex Analysis Physics Notes | EduRev(6.13) 

Now let A,and be the matrices whose ijth elements are∂xi/∂yj,∂yi/∂zj and∂xi/∂zj respectively. We can then write (6.13) as the matrix product Jacobian, Complex Analysis Physics Notes | EduRev (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 Jacobian, Complex Analysis Physics Notes | EduRev (6.15) 

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

Jxz=Jxy Jyz 

or, in the usual notation,

Jacobian, Complex Analysis Physics Notes | EduRev (6.16) 

As a special case, if the set zi is taken to be identical to the set xi,and the obvious result Jxx= 1 is used, we obtain 

Jxy Jyx=1 

or, in the usual notation, 

Jacobian, Complex Analysis Physics Notes | EduRev (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|=|AT|,where AT is the transpose of A, we can interchange the rows and columns in the determinantal form of the Jacobian without changing its value. 

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