Ratio of Independent Gammas Yields a Beta Distribution

Let $X \sim \operatorname{Gamma}(\alpha, 1)$ and $Y \sim \operatorname{Gamma}(\beta, 1)$ be independent. Using the transformation $(W, S) = \bigl(X/(X+Y),\; X+Y\bigr)$:

(a) Compute the Jacobian of the inverse map.

(b) Derive the joint density $f_{W,S}$ and marginalize to show $W \sim \operatorname{Beta}(\alpha, \beta)$.

(c) Show that $W$ and $S$ are independent.