Question

In Exercises \(27-30,\) find the Jacobian \(\partial(x, y, z) / \partial(u, v, w)\) for the indicated change of variables. If \(x=f(u, v, w), y=g(u, v, w)\) and \(z=h(u, v, w),\) then the Jacobian of \(x, y,\) and \(z\) with respect to \(u, v,\) and \(w\) is \(\frac{\partial(x, y, z)}{\partial(u, v, w)}=\left|\begin{array}{lll}\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w}\end{array}\right| .\) Spherical Coordinates $$ x=\rho \sin \phi \cos \theta, y=\rho \sin \phi \sin \theta, z=\rho \cos \phi $$

Short answer

The Jacobian of the spherical coordinates which are represented by \(x=\rho \sin \phi \cos \theta\), \(y=\rho \sin \phi \sin \theta\), \(z=\rho \cos \phi\) with respect to \(\rho\), \(\phi\), and \(\theta\) is therefore the determinant of the matrix formed by the partial derivatives, as calculated in the steps above.

Step-by-step solution

Compute the Partial Derivatives

Firstly, calculate partial derivatives of each function with respect to each independent variable (u,v,w) which in this case are \(\rho\) , \(\phi\) , and \(\theta\). For \(x=\rho \sin \phi \cos \theta\), the following are the partial derivatives: \(\frac{\partial x}{\partial \rho} = \sin (\phi) \cos(\theta)\), \(\frac{\partial x}{\partial \phi} = \rho \cos (\phi) \cos(\theta)\), \(\frac{\partial x}{\partial \theta} = -\rho\sin (\phi) \sin(\theta)\), and similarly for y and z.

Arrange the Partial Derivatives into a matrix

Place the calculated derivatives into a 3x3 grid (which represents a determinant). This matrix should be set out as follows: \(\frac{\partial(x, y, z)}{\partial(\rho, \phi, \theta)}=\left|\begin{array}{lll}\frac{\partial x}{\partial \rho} & \frac{\partial x}{\partial \phi} & \frac{\partial x}{\partial \theta} \ \frac{\partial y}{\partial \rho} & \frac{\partial y}{\partial \phi} & \frac{\partial y}{\partial \theta} \ \frac{\partial z}{\partial \rho} & \frac{\partial z}{\partial \phi} & \frac{\partial z}{\partial \theta}\end{array}\right|\)

Evaluate the Determinant

The final step involves calculating the determinant of the matrix. The determinant of a 3x3 matrix can be evaluated using cofactor expansion method. Remember that the sign changes for each element as you move across the rows and columns.