1. Find the Jacobian of the following transformations. (a) \( x=u^{2}+u v, \quad y=u v^{2} \)

To find the Jacobian of the given transformations, we need to calculate the partial derivatives of \( x \) and \( y \) with respect to \( u \) and \( v \). Given transformations: \[ x = u^2 + uv \] \[ y = uv^2 \] Let's calculate the partial derivatives: 1. Partial derivative of \( x \) with respect to \( u \): \[ \frac{\partial x}{\partial u} = 2u + v \] 2. Partial derivative of \( x \) with respect to \( v \): \[ \frac{\partial x}{\partial v} = u \] 3. Partial derivative of \( y \) with respect to \( u \): \[ \frac{\partial y}{\partial u} = v^2 \] 4. Partial derivative of \( y \) with respect to \( v \): \[ \frac{\partial y}{\partial v} = 2uv \] Now, we can calculate the Jacobian matrix using these partial derivatives: \[ J = \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix} = \begin{bmatrix} 2u + v & u \\ v^2 & 2uv \end{bmatrix} \] Therefore, the Jacobian of the given transformations is: \[ J = \begin{bmatrix} 2u + v & u \\ v^2 & 2uv \end{bmatrix} \]