Beta Distribution and the Beta Function from Independent Gammas

Let $X \sim \text{Gamma}(\alpha, 1)$ and $Y \sim \text{Gamma}(\beta, 1)$ be independent.

(a) Define $U = \frac{X}{X+Y}$ and $V = X + Y$. Compute the Jacobian of the transformation $(X, Y) \mapsto (U, V)$.

(b) Derive the joint PDF of $(U, V)$ and show that $U$ and $V$ are independent.

(c) Identify the marginal distribution of $U$ and prove that $B(\alpha, \beta) = \frac{\Gamma(\alpha)\,\Gamma(\beta)}{\Gamma(\alpha+\beta)}$, where $B(\alpha,\beta) = \int_0^1 u^{\alpha-1}(1-u)^{\beta-1}\,du$.