We are asked to compute the Jacobian determinant of the transformation from cylindrical coordinates $(r, \theta, z)$ to Cartesian coordinates $(x, y, z)$.

Given the coordinate transformations:

• $x = r \cos(\theta)$
• $y = r \sin(\theta)$
• $z = z$

We want to find the determinant of the Jacobian matrix $J$, which is defined as: $J = \frac{\partial(x, y, z)}{\partial(r, \theta, z)}$

Step 1: Compute the partial derivatives

The Jacobian matrix $J$ is formed by taking the partial derivatives of $x$, $y$, and $z$ with respect to $r$, $\theta$, and $z$:

\frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} & \frac{\partial x}{\partial z} \\ \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} & \frac{\partial y}{\partial z} \\ \frac{\partial z}{\partial r} & \frac{\partial z}{\partial \theta} & \frac{\partial z}{\partial z} \end{pmatrix}$$ Let's compute each of these partial derivatives: - $$\frac{\partial x}{\partial r} = \cos(\theta)$$ - $$\frac{\partial x}{\partial \theta} = -r \sin(\theta)$$ - $$\frac{\partial x}{\partial z} = 0$$ - $$\frac{\partial y}{\partial r} = \sin(\theta)$$ - $$\frac{\partial y}{\partial \theta} = r \cos(\theta)$$ - $$\frac{\partial y}{\partial z} = 0$$ - $$\frac{\partial z}{\partial r} = 0$$ - $$\frac{\partial z}{\partial \theta} = 0$$ - $$\frac{\partial z}{\partial z} = 1$$ Thus, the Jacobian matrix $$J$$ becomes: $$J = \begin{pmatrix} \cos(\theta) & -r \sin(\theta) & 0 \\ \sin(\theta) & r \cos(\theta) & 0 \\ 0 & 0 & 1 \end{pmatrix}$$ ### Step 2: Compute the determinant of the Jacobian matrix The determinant of this 3x3 matrix is: $$\text{det}(J) = \begin{vmatrix} \cos(\theta) & -r \sin(\theta) & 0 \\ \sin(\theta) & r \cos(\theta) & 0 \\ 0 & 0 & 1 \end{vmatrix}$$ This determinant can be computed by expanding along the third column (since it has a lot of zeros): $$\text{det}(J) = 1 \cdot \begin{vmatrix} \cos(\theta) & -r \sin(\theta) \\ \sin(\theta) & r \cos(\theta) \end{vmatrix}$$ Now, calculate the 2x2 determinant: $$\begin{vmatrix} \cos(\theta) & -r \sin(\theta) \\ \sin(\theta) & r \cos(\theta) \end{vmatrix} = r \left(\cos^2(\theta) + \sin^2(\theta)\right) = r$$ (since $$\cos^2(\theta) + \sin^2(\theta) = 1$$). ### Final Answer Thus, the Jacobian determinant is: $$\boxed{r}$$ This result means that the volume element in cylindrical coordinates is scaled by a factor of $$r$$ relative to the volume element in Cartesian coordinates. Would you like further details or clarifications? Here are some related questions: 1. How do we apply Jacobians to coordinate transformations in different dimensions? 2. What does the Jacobian determinant represent physically in terms of volume scaling? 3. How is the Jacobian matrix used in converting integrals between coordinate systems? 4. How does the cylindrical coordinate system differ from spherical coordinates? 5. What are the key properties of the determinant of a matrix? **Tip:** The Jacobian determinant tells us how volume changes under a transformation between coordinate systems.