Question

Evaluate the Jacobian for the transformation from spherical to rectangular coordinates: \(x=\rho \sin \varphi \cos \theta, y=\rho \sin \varphi \sin \theta, z=\rho \cos \varphi .\) Show that \(J(\rho, \varphi, \theta)=\rho^{2} \sin \varphi\)

Short answer

Answer: The Jacobian determinant is $J(\rho, \varphi, \theta) = \rho^2 \sin \varphi$.

Step-by-step solution

Calculate the partial derivatives

First, we need to find the partial derivatives of \(x, y, z\) with respect to \(\rho, \varphi, \theta\). This will give us the elements of the Jacobian matrix. - \(\frac{\partial x}{\partial \rho} = \sin \varphi \cos \theta\) - \(\frac{\partial x}{\partial \varphi} = \rho \cos \varphi \cos \theta\) - \(\frac{\partial x}{\partial \theta} = -\rho \sin \varphi \sin \theta\) - \(\frac{\partial y}{\partial \rho} = \sin \varphi \sin \theta\) - \(\frac{\partial y}{\partial \varphi} = \rho \cos \varphi \sin \theta\) - \(\frac{\partial y}{\partial \theta} = \rho \sin \varphi \cos \theta\) - \(\frac{\partial z}{\partial \rho} = \cos \varphi\) - \(\frac{\partial z}{\partial \varphi} = -\rho \sin \varphi\) - \(\frac{\partial z}{\partial \theta} = 0\)

Construct the Jacobian matrix and find its determinant

Now that we have all the partial derivatives, we can construct the Jacobian matrix: $$ \begin{pmatrix} \frac{\partial x}{\partial \rho} & \frac{\partial x}{\partial \varphi} & \frac{\partial x}{\partial \theta} \\ \frac{\partial y}{\partial \rho} & \frac{\partial y}{\partial \varphi} & \frac{\partial y}{\partial \theta} \\ \frac{\partial z}{\partial \rho} & \frac{\partial z}{\partial \varphi} & \frac{\partial z}{\partial \theta} \end{pmatrix} = \begin{pmatrix} \sin \varphi \cos \theta & \rho \cos \varphi \cos \theta & -\rho \sin \varphi \sin \theta \\ \sin \varphi \sin \theta & \rho \cos \varphi \sin \theta & \rho \sin \varphi \cos \theta \\ \cos \varphi & -\rho \sin \varphi & 0 \end{pmatrix} $$ Now we must find the determinant of this matrix to obtain the Jacobian determinant \(J(\rho, \varphi, \theta)\). Using the definition of a determinant, we get: $$ \begin{aligned} J(\rho, \varphi, \theta) &= (\sin \varphi \cos \theta) \begin{vmatrix} \rho \cos \varphi \sin \theta & \rho \sin \varphi \cos \theta \\ -\rho \sin \varphi & 0 \end{vmatrix} - (\rho \cos \varphi \cos \theta) \begin{vmatrix} \sin \varphi \sin \theta & \rho \sin \varphi \cos \theta \\ \cos \varphi & 0 \end{vmatrix} + (-\rho \sin \varphi \sin \theta) \begin{vmatrix} \sin \varphi \sin \theta & \rho \cos \varphi \sin \theta \\ \cos \varphi & -\rho \sin \varphi \end{vmatrix} \\ &= (\sin \varphi \cos \theta) [(0)-(-\rho \sin \varphi)(\rho \sin \varphi \cos \theta)] - (\rho \cos \varphi \cos \theta) [(0)-(\cos \varphi)(\rho \sin \varphi \cos \theta)]. \end{aligned} $$ Simplifying, we get: $$ J(\rho, \varphi, \theta) = (\sin \varphi \cos \theta) [\rho^2 \sin^2 \varphi \cos \theta] $$ Further simplifying, we obtain the final answer: $$ J(\rho, \varphi, \theta) = \rho^2 \sin \varphi $$

Conclusion

We have shown that the Jacobian determinant for the transformation from spherical to rectangular coordinates is \(J(\rho, \varphi, \theta) = \rho^2 \sin \varphi\).

Key concepts

Understanding Spherical Coordinates

Spherical coordinates are a three-dimensional way of defining points in space. They are especially useful in scenarios involving spheres or radial systems, such as astronomy or electromagnetics. Instead of using linear measures, spherical coordinates use:

• \(\rho\) (rho): The radial distance from the origin to the point.
• \(\varphi\) (phi): The polar angle measured from the positive z-axis, similar to latitude.
• \(\theta\) (theta): The azimuthal angle in the xy-plane from the positive x-axis, akin to longitude.

This system is helpful because many natural systems have spherical symmetry, meaning calculations using spherical coordinates can be simpler than using traditional Cartesian coordinates.
For transformations, understanding these angles and how they relate to their rectangular equivalents is crucial.

Converting to Rectangular Coordinates

Rectangular, or Cartesian coordinates, describe a point in space using x, y, and z values, representing horizontal, vertical, and depth dimensions respectively. Understanding how to convert between spherical and rectangular coordinates is essential in many fields.
The transformation equations are:

• x = \(\rho \sin \varphi \cos \theta\): This equation defines the x-coordinate using the spherical coordinates.
• y = \(\rho \sin \varphi \sin \theta\): Similarly, this describes the y-coordinate.
• z = \(\rho \cos \varphi\): This equation assigns the z-coordinate.

These equations help transition calculations from a spherical to a rectangular system. By mastering these transformations, one can effectively work across different coordinate systems.

The Role of Partial Derivatives

Partial derivatives are a fundamental concept in calculus, representing the rate of change of a function concerning one variable, keeping the others constant. They are vital when dealing with multi-variable functions, like those in physics and engineering.In this context, partial derivatives help form the Jacobian matrix.
Each element in the Jacobian matrix is a partial derivative of the transformation equations, giving insight into how each coordinate relationship changes:

• The derivative \(\frac{\partial x}{\partial \rho}\), for example, shows how x changes with respect to \(\rho\).
• This process repeats for \(\varphi\) and \(\theta\) too, constructing a complete picture with the Jacobian matrix.

The determinant of this matrix, known as the Jacobian determinant, helps evaluate how volume or area transforms under spherical to rectangular conversion. It's especially important in physics and engineering where transformation behavior matters.