theorem transformation rule let be an -dimensional vector of continuous random variables with pdf let ( )where is an invertible mapping with inverse - and matrix of jacobi that isthe matrix of partial derivatives of thenat (xthe random vector has pdf (zf (xf - ( )det( - ( ))| rn det( ( ))( prooffor fixed xlet ( )and thus - (zin the neighborhood of xthe function behaves like linear functionin the sense that ( dg(xj (xd for small vectors dsee also section consequentlyan infinitesimally small -dimensional rectangle at with volume is transformed into an infinitesimally small -dimensional parallelepiped at with volume det( ( ))nowas in the proof of the linear caselet be small cube around (xwith volume hn let be the image of under - thenhn (zp[ chn det( - ( )) ( )and since det( - ( )) /det( ( ))|( follows as goes to typicallyin coordinate transformations it is - that is given -that isan expression for as function of example (polar transformsuppose xy are independent and have standard normal distribution the joint pdf is fx, (xy ( + (xyr in polar coordinates we have cos th and sin th( where is the radius and th [ pthe angle of the point (xywhat is the joint pdf of and thby the radial symmetry of the bivariate normal distributionwe would expect th to be uniform on ( pbut what is the pdf of rto work out the joint pdfconsider the inverse transformation - defined by - cos th -th sin th the corresponding matrix of jacobi is cos th jg- (rthsin th - sin th cos th