5) If \( u=2 x y, v=x^{2}-y^{2}, x=r \cos \theta \) and \( y=r \sin x \)

It appears you're working with a coordinate transformation problem involving the variables \( u \) and \( v \) defined in terms of Cartesian coordinates \( x \) and \( y \), and expressing \( x \) and \( y \) in polar coordinates \( r \) and \( \theta \). Let's carefully break down the given information and proceed step-by-step. ## Given: 1. **Transformation Equations:** \[ u = 2xy \quad \text{and} \quad v = x^{2} - y^{2} \] 2. **Polar Coordinates Representation:** \[ x = r \cos \theta \quad \text{and} \quad y = r \sin \theta \] *(Note: It seems there was a typo in your original message where \( y \) was expressed as \( y = r \sin x \). Based on standard polar coordinates, it should be \( y = r \sin \theta \).)* ## Objective: Typically, such problems aim to **express \( u \) and \( v \) in terms of polar coordinates \( r \) and \( \theta \)** and possibly to **find the Jacobian determinant** for the transformation. We'll proceed with these objectives. ### 1. Express \( u \) and \( v \) in Terms of \( r \) and \( \theta \): Substitute \( x = r \cos \theta \) and \( y = r \sin \theta \) into the transformation equations. - **For \( u \):** \[ u = 2xy = 2(r \cos \theta)(r \sin \theta) = 2r^{2} \sin \theta \cos \theta \] Using the double-angle identity \( \sin 2\theta = 2 \sin \theta \cos \theta \): \[ u = r^{2} \sin 2\theta \] - **For \( v \):** \[ v = x^{2} - y^{2} = (r \cos \theta)^{2} - (r \sin \theta)^{2} = r^{2} (\cos^{2} \theta - \sin^{2} \theta) \] Using the double-angle identity \( \cos 2\theta = \cos^{2} \theta - \sin^{2} \theta \): \[ v = r^{2} \cos 2\theta \] ### 2. Finding the Jacobian Determinant: The Jacobian determinant is essential when changing variables in multiple integrals. It determines how area (or volume) scales under the transformation. **Jacobian Matrix:** \[ J = \begin{bmatrix} \frac{\partial u}{\partial r} & \frac{\partial u}{\partial \theta} \\ \frac{\partial v}{\partial r} & \frac{\partial v}{\partial \theta} \end{bmatrix} \] **Compute the Partial Derivatives:** - **For \( u = r^{2} \sin 2\theta \):** \[ \frac{\partial u}{\partial r} = 2r \sin 2\theta \] \[ \frac{\partial u}{\partial \theta} = r^{2} \cdot 2 \cos 2\theta = 2r^{2} \cos 2\theta \] - **For \( v = r^{2} \cos 2\theta \):** \[ \frac{\partial v}{\partial r} = 2r \cos 2\theta \] \[ \frac{\partial v}{\partial \theta} = r^{2} \cdot (-2 \sin 2\theta) = -2r^{2} \sin 2\theta \] **Jacobian Matrix Filled In:** \[ J = \begin{bmatrix} 2r \sin 2\theta & 2r^{2} \cos 2\theta \\ 2r \cos 2\theta & -2r^{2} \sin 2\theta \end{bmatrix} \] **Compute the Determinant:** \[ \text{det}(J) = (2r \sin 2\theta)(-2r^{2} \sin 2\theta) - (2r^{2} \cos 2\theta)(2r \cos 2\theta) \] \[ = -4r^{3} \sin^2 2\theta - 4r^{3} \cos^2 2\theta \] \[ = -4r^{3} (\sin^2 2\theta + \cos^2 2\theta) \] \[ = -4r^{3} (1) = -4r^{3} \] **Absolute Value of the Jacobian:** \[ |\text{det}(J)| = 4r^{3} \] *(The negative sign indicates orientation change, but when dealing with areas or volumes, we typically use the absolute value.)* ## Summary: - **Transformed Equations:** \[ u = r^{2} \sin 2\theta \quad \text{and} \quad v = r^{2} \cos 2\theta \] - **Jacobian Determinant:** \[ |\text{det}(J)| = 4r^{3} \] ## Applications: 1. **Change of Variables in Integration:** When performing double integrals, the Jacobian determinant adjusts the area element: \[ dA = | \text{det}(J) | \, dr \, d\theta = 4r^{3} \, dr \, d\theta \] 2. **Analyzing Transformation Properties:** Knowing the Jacobian helps understand how the transformation scales areas, which is crucial in physics and engineering applications. If your problem involves further steps or specific applications of this transformation, please provide additional details so I can assist you more effectively!