> [!NOTE] Theorem (Change of Variables Formula for Double Integral) > Let $\underline{F}:U \subseteq \mathbb{R}^{2}\to \mathbb{R}^{2}$ be a [[Bijection|bijection]] that is continuously differentiable. If $f: F(S) \to \mathbb{R}$ is twice-[[Darboux Integrable Function|integrable]], then $\int \int_{F(S)} f(x,y)\, dx\, dy = \int \int_{S} f (\underline{F}(u,v)) \, \lvert \det D\mathbf{F}(u,v) \rvert \, du\, dv $where the $\det D\mathbf{F}(u,v)$, denotes the [[Jacobian|Jacobian]] and $D\mathbf{F}(u,v)$ the matrix of [[Directional Derivative of Real-Valued Function of Several Real Variables|directional derivatives]]: $D\mathbf{F} = \begin{pmatrix}x_{u} & x_{v } \\y_{u} & y_{v} \end{pmatrix}$ **Proof**: ...