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I am studying multivariate statistics and I don't understand the meaning of Jacobian of the transformation for pdf of function of random vectors.

If I have a random vector, let's say bivariate, (X,Y) with joint density f(X,Y) I know how to find the joint pdf of (U,V) with U = g(X,Y) and V = h(X,Y) I did a lot of exercises and I can easily calculate joint pfd of functions of random vectors.

The formula for joint pdf of functions of random vectors involves the determinant of the Jacobian matrix of inverse functions. In the univariate case I understand that I have a derivative because the pdf is the derivative of the CDF. In the multivariate case why the determinant of the Jacobian?

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    $\begingroup$ It might come down to just what you mean by "understand" and "meaning." One level of understanding and meaning (algebraic and abstract) is described in my answer at stats.stackexchange.com/questions/36093. Another (geometric and concrete) is given at stats.stackexchange.com/a/4223/919. Although that one is in one dimension, nothing changes in higher dimensions: the Jacobian tells you how to convert the units of length, area, volume, or (generally) hypervolume. What, then, are you asking for? To understand densities, units of measure, Jacobians, determinants, or something else? $\endgroup$ – whuber Mar 30 '17 at 14:28

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