1. **Problem statement:** We need to find the Jacobian determinants for two cases: (i) Given $x=12(u^2 - v^2)$ and $y=uv$, compute $\frac{\partial(F,G)}{\partial(u,v)}$ where $F=x$ and $G=y$. (ii) Given $x=r\cos\theta$ and $y=r\sin\theta$, find the Jacobian $J=\frac{\partial(u,v)}{\partial(x,y)}$ or equivalently $\frac{\partial(x,y)}{\partial(u,v)}$. 2. **Formula and rules:** The Jacobian determinant of functions $F(u,v)$ and $G(u,v)$ is: $$ J = \frac{\partial(F,G)}{\partial(u,v)} = \begin{vmatrix} \frac{\partial F}{\partial u} & \frac{\partial F}{\partial v} \\ \frac{\partial G}{\partial u} & \frac{\partial G}{\partial v} \end{vmatrix} = \frac{\partial F}{\partial u} \frac{\partial G}{\partial v} - \frac{\partial F}{\partial v} \frac{\partial G}{\partial u} $$ This measures how the area changes under the transformation. 3. **Part (i): Compute $\frac{\partial(F,G)}{\partial(u,v)}$ for $F=x=12(u^2 - v^2)$ and $G=y=uv$** - Compute partial derivatives: - $\frac{\partial x}{\partial u} = 12 \cdot 2u = 24u$ - $\frac{\partial x}{\partial v} = 12 \cdot (-2v) = -24v$ - $\frac{\partial y}{\partial u} = v$ - $\frac{\partial y}{\partial v} = u$ - Jacobian determinant: $$ \frac{\partial(x,y)}{\partial(u,v)} = (24u)(u) - (-24v)(v) = 24u^2 + 24v^2 = 24(u^2 + v^2) $$ 4. **Part (ii): Find $J=\frac{\partial(u,v)}{\partial(x,y)}$ if $x=r\cos\theta$, $y=r\sin\theta$** - Here, $u=r$, $v=\theta$ and variables are $(r,\theta)$ and $(x,y)$. - Compute Jacobian $\frac{\partial(x,y)}{\partial(r,\theta)}$: - $\frac{\partial x}{\partial r} = \cos\theta$ - $\frac{\partial x}{\partial \theta} = -r \sin\theta$ - $\frac{\partial y}{\partial r} = \sin\theta$ - $\frac{\partial y}{\partial \theta} = r \cos\theta$ - Jacobian determinant: $$ \frac{\partial(x,y)}{\partial(r,\theta)} = \cos\theta \cdot r \cos\theta - (-r \sin\theta) \cdot \sin\theta = r(\cos^2\theta + \sin^2\theta) = r $$ - Since $J=\frac{\partial(u,v)}{\partial(x,y)}$ is the inverse of $\frac{\partial(x,y)}{\partial(u,v)}$, we have: $$ J = \frac{\partial(r,\theta)}{\partial(x,y)} = \frac{1}{r} $$ **Final answers:** - (i) $\frac{\partial(x,y)}{\partial(u,v)} = 24(u^2 + v^2)$ - (ii) $J = \frac{\partial(r,\theta)}{\partial(x,y)} = \frac{1}{r}$