To compute the Jacobian of the spherical coordinate map, let’s first define the transformation from spherical coordinates $(\rho, \theta, \phi)$ to Cartesian coordinates $(x, y, z)$:

$x = \rho \sin \phi \cos \theta$ $y = \rho \sin \phi \sin \theta$ $z = \rho \cos \phi$

where:

• $\rho$ is the radial distance,
• $\theta$ is the azimuthal angle (longitude),
• $\phi$ is the polar angle (colatitude).

Step 1: Compute Partial Derivatives

To find the Jacobian, we need to compute the partial derivatives of $x$, $y$, and $z$ with respect to $\rho$, $\theta$, and $\phi$.

1. Partial derivatives of $x$: $\frac{\partial x}{\partial \rho} = \sin \phi \cos \theta$ $\frac{\partial x}{\partial \theta} = -\rho \sin \phi \sin \theta$ $\frac{\partial x}{\partial \phi} = \rho \cos \phi \cos \theta$

2. Partial derivatives of $y$: $\frac{\partial y}{\partial \rho} = \sin \phi \sin \theta$ $\frac{\partial y}{\partial \theta} = \rho \sin \phi \cos \theta$ $\frac{\partial y}{\partial \phi} = \rho \cos \phi \sin \theta$

3. Partial derivatives of $z$: $\frac{\partial z}{\partial \rho} = \cos \phi$ $\frac{\partial z}{\partial \theta} = 0$ $\frac{\partial z}{\partial \phi} = -\rho \sin \phi$

Step 2: Form the Jacobian Matrix

Now, we arrange these partial derivatives into the Jacobian matrix $J$:

\frac{\partial x}{\partial \rho} & \frac{\partial x}{\partial \theta} & \frac{\partial x}{\partial \phi} \\ \frac{\partial y}{\partial \rho} & \frac{\partial y}{\partial \theta} & \frac{\partial y}{\partial \phi} \\ \frac{\partial z}{\partial \rho} & \frac{\partial z}{\partial \theta} & \frac{\partial z}{\partial \phi} \end{bmatrix} = \begin{bmatrix} \sin \phi \cos \theta & -\rho \sin \phi \sin \theta & \rho \cos \phi \cos \theta \\ \sin \phi \sin \theta & \rho \sin \phi \cos \theta & \rho \cos \phi \sin \theta \\ \cos \phi & 0 & -\rho \sin \phi \end{bmatrix}$$ ### Step 3: Compute the Determinant of the Jacobian The Jacobian determinant $$J$$ represents the volume scaling factor for the transformation from spherical to Cartesian coordinates. Computing this determinant, we find: $$\det(J) = \rho^2 \sin \phi$$ ### Result The Jacobian of the spherical coordinate map is: $$\det(J) = \rho^2 \sin \phi$$ This factor $$\rho^2 \sin \phi$$ appears in volume integrals when converting from Cartesian to spherical coordinates. Would you like more details on the steps involved, or have any questions? ### Related Questions 1. How does the Jacobian help in transforming integrals to spherical coordinates? 2. What does the Jacobian determinant signify geometrically? 3. How would you apply this Jacobian in a triple integral setup? 4. Can this Jacobian be applied to cylindrical coordinates? 5. What are common applications of spherical coordinates in physics? ### Tip When working with Jacobians, always verify each partial derivative carefully, as small errors can lead to incorrect transformations.