Let $X$ ~ $N(\mu_X, \sigma_X^2)$ and $Y$ ~ $N(\mu_Y, \sigma_Y^2)$ be normal independent random variables and let $Z = X + Y$. Find the distribution of $Z$ using the theorem of density transformation of random vector.
Theorem: Let random vector $X = (X_1, ..., X_n)^T$ have density $p(x)$ in consideration of Lebesgue's measure in $R_n$. Let $t$ be the mapping from $R_n$ to $R_n$, which is regular and injective on open set $G$, for which we have $\int_Gp(x)dx = 1$. Let $\tau$ be the inverse of $t:G \rightarrow t(G)$. Then the random vector $Y = t(X)$ has a density in consideration of Lebesgue's measure and the density is:
$g(y) = p(\tau(y))|D_{\tau}(y)|, y \in t(G)$
$g(y) = 0, y \notin t(G)$.
They don't say what is $D_{\tau}$, but I guess it is determinant of something.