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

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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:
$$ J = \begin{bmatrix} 2u + v & u \ v^2 & 2uv \end{bmatrix} $$
The Jacobian of the transformations is: $$ J = \begin{bmatrix} 2u + v & u \ v^2 & 2uv \end{bmatrix} $$

Find the Jacobian of the following transformations.
(a)

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