Say that $X \in \mathbb{R}^n$ is a vector of $n$ r.v.'s with pdf $p(x_1,\ldots,x_n)$. Let's consider now the linear map $Y = A X$ where $Y \in \mathbb{R}^m$ with $m < n$. I am seeking $p(y_1,\ldots,y_m)$. The case where $n=m$ is treated in all elementary probability textbooks (change of variables formula) however I could not find much about what happens when $m < n$ since in this case the Jacobian is not invertible. I am interested both in the case where $X$ is normally distributed and the case where is $p(x_1,\ldots,x_n)$ is a generic continuous density. In my specific problem the map $A$ is an orthogonal projection to a lower dimensional subspace ($m = n-1$).

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    $\begingroup$ By far the easiest way (IMO) to deal with this uses differential algebra, as I describe (very generally) at stats.stackexchange.com/a/154298/919. However, this isn't needed in your case, because orthogonal projection is equivalent to computing the marginal distribution when you choose appropriately adapted coordinates for the calculation: In short, you simply integrate. $\endgroup$ – whuber Dec 31 '18 at 20:58
  • $\begingroup$ @whuber thanks for your answer. Do you know any "modern" textbook on differential algebra treating these kind of applications? $\endgroup$ – Arrigo Benedetti Dec 31 '18 at 22:44
  • $\begingroup$ The most elementary and rigorous one I know, and reference often, is Michael Spivak's Calculus on Manifolds. Although it was written 50 years ago, it is still fairly modern. I don't know of any textbook devoted to illustrating applications of differential algebra to statistical problems, though. $\endgroup$ – whuber Dec 31 '18 at 22:51

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