To solve this problem, we need to calculate the **Jacobian matrix** \( J \), which consists of the partial derivatives of the given parametric equations \( x(u,v) = 4\cos(v) \) and \( y(u,v) = u\sin(v) \). This matrix allows us to understand how changes in the variables \( u \) and \( v \) affect the coordinates \( x \) and \( y \).

The Jacobian matrix \( J \) is defined as:

\[
J = \begin{pmatrix}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\
\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v}
\end{pmatrix}
\]

### Step 1: Compute the partial derivatives
1. **Partial derivative of \( x \) with respect to \( u \)**:
\[
\frac{\partial x}{\partial u} = \frac{\partial}{\partial u}(4\cos(v)) = 0
\]

2. **Partial derivative of \( x \) with respect to \( v \)**:
\[
\frac{\partial x}{\partial v} = \frac{\partial}{\partial v}(4\cos(v)) = -4\sin(v)
\]

3. **Partial derivative of \( y \) with respect to \( u \)**:
\[
\frac{\partial y}{\partial u} = \frac{\partial}{\partial u}(u\sin(v)) = \sin(v)
\]

4. **Partial derivative of \( y \) with respect to \( v \)**:
\[
\frac{\partial y}{\partial v} = \frac{\partial}{\partial v}(u\sin(v)) = u\cos(v)
\]

### Step 2: Construct the Jacobian matrix
Now we can construct the Jacobian matrix \( J \):

\[
J = \begin{pmatrix}
0 & -4\sin(v) \\
\sin(v) & u\cos(v)
\end{pmatrix}
\]

### Step 3: Find \( J_1 \) and \( J_2 \)
We assume \( J_1 \) and \( J_2 \) refer to the first and second columns of the Jacobian matrix, respectively:

- \( J_1 = \begin{pmatrix} 0 \\ \sin(v) \end{pmatrix} \)
- \( J_2 = \begin{pmatrix} -4\sin(v) \\ u\cos(v) \end{pmatrix} \)

These represent the rates of change of the coordinates with respect to \( u \) and \( v \).

Would you like further clarification or have additional questions?

Here are 5 questions to further explore this topic:
1. What is the interpretation of the Jacobian determinant for this system?
2. How does the Jacobian matrix relate to transformations between coordinate systems?
3. What is the geometric significance of the columns \( J_1 \) and \( J_2 \)?
4. How do changes in \( u \) and \( v \) affect the resulting shape in \( xy \)-coordinates?
5. How can you use the Jacobian matrix to check if the transformation is invertible?

**Tip:** The Jacobian matrix helps in transforming between different coordinate systems and is crucial in multivariable calculus for understanding area and volume transformations.

Understand how to find the Jacobian matrix for the parametric equations x = 4cos(v) and y = u sin(v). This detailed solution covers the steps to compute the partial derivatives and construct the Jacobian matrix, including how to find J1 and J2. Suitable for students studying multivariable calculus at the college level.

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