We are given the parametric equations:

$x = 4 \cos v$ $y = u \sin v$

We are tasked with showing that $J_1 \times J_2 = 1$, which suggests that $J_1$ and $J_2$ might represent the partial derivatives of $x$ and $y$ with respect to the variables $u$ and $v$. This type of equation often corresponds to the Jacobian determinant of the transformation from $(u, v)$ to $(x, y)$.

Step 1: Partial derivatives

We'll compute the partial derivatives of $x$ and $y$ with respect to $u$ and $v$.

• $x = 4 \cos v$

  • $\frac{\partial x}{\partial u} = 0$ (since $x$ does not depend on $u$)
  • $\frac{\partial x}{\partial v} = -4 \sin v$

• $y = u \sin v$

  • $\frac{\partial y}{\partial u} = \sin v$
  • $\frac{\partial y}{\partial v} = u \cos v$

Step 2: Jacobian matrix

The Jacobian matrix $J$ for the transformation is:

\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{pmatrix} = \begin{pmatrix} 0 & -4 \sin v \\ \sin v & u \cos v \end{pmatrix}$$ ### Step 3: Determinant of the Jacobian The determinant of the Jacobian matrix is: $$\det(J) = \left( 0 \times u \cos v \right) - \left( -4 \sin v \times \sin v \right) = 0 + 4 \sin^2 v$$ $$\det(J) = 4 \sin^2 v$$ ### Step 4: Condition for $$J_1 \times J_2 = 1$$ For the determinant to be equal to 1, we require that: $$4 \sin^2 v = 1$$ Solving for $$\sin^2 v$$, we get: $$\sin^2 v = \frac{1}{4}$$ Taking the square root: $$\sin v = \pm \frac{1}{2}$$ Thus, for $$J_1 \times J_2 = 1$$, $$v$$ must satisfy $$\sin v = \pm \frac{1}{2}$$. This occurs for specific values of $$v$$, such as $$v = \frac{\pi}{6}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{11\pi}{6}, \ldots$$. ### Conclusion Under the condition that $$\sin v = \pm \frac{1}{2}$$, we have $$J_1 \times J_2 = 1$$, which completes the proof. Would you like to explore any step further or have additional questions? Here are 5 related questions: 1. What are the general properties of Jacobians in transformations? 2. How does the Jacobian determinant relate to area scaling in transformations? 3. Can we generalize this result for different parametric forms of $$x$$ and $$y$$? 4. What are the real-world applications of computing Jacobians in transformations? 5. How does this concept extend to higher-dimensional transformations? **Tip:** Jacobians are often used to convert between coordinate systems, such as from polar to Cartesian coordinates.