Ex.9 Compute the Jacobian of the spherical coordinate map.

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 \(J\):

\[
J = \begin{bmatrix}
\frac{\partial x}{\partial \rho} & \frac{\partial x}{\partial \theta} & \frac{\partial x}{\partial \phi} \\
\frac{\partial y}{\partial \rho} & \frac{\partial y}{\partial \theta} & \frac{\partial y}{\partial \phi} \\
\frac{\partial z}{\partial \rho} & \frac{\partial z}{\partial \theta} & \frac{\partial z}{\partial \phi}
\end{bmatrix} = \begin{bmatrix}
\sin \phi \cos \theta & -\rho \sin \phi \sin \theta & \rho \cos \phi \cos \theta \\
\sin \phi \sin \theta & \rho \sin \phi \cos \theta & \rho \cos \phi \sin \theta \\
\cos \phi & 0 & -\rho \sin \phi
\end{bmatrix}
\]

### 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.

Compute the Jacobian of the spherical coordinate map.

Learn how to compute the Jacobian of the spherical coordinate map and its determinant. This detailed solution covers the transformation equations from spherical to Cartesian coordinates, the computation of partial derivatives, and the interpretation of the Jacobian determinant as a volume scaling factor.

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