Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the Jacobian for the transformation from rectangular coordinates to spherical coordinates.

Step-by-Step Solution

Answer

The Jacobian for the transformation is \(\rho^2 \sin \phi\).

Step 1: Understand the Relationship between Coordinate Systems

In rectangular coordinates (Cartesian), a point in space is given by \((x, y, z)\). In spherical coordinates, the same point is given by \((\rho, \theta, \phi)\), where \(\rho\) is the radial distance from the origin, \(\theta\) is the azimuthal angle in the \(xy\)-plane from the positive \(x\)-axis, and \(\phi\) is the polar angle from the positive \(z\)-axis. The transformation equations are:\[ x = \rho \sin \phi \cos \theta \] \[ y = \rho \sin \phi \sin \theta \] \[ z = \rho \cos \phi \].

Step 2: Set Up the Jacobian Matrix

The Jacobian matrix for a transformation from rectangular to spherical coordinates is constructed using the partial derivatives of the transformations. Specifically, it is a \(3 \times 3\) matrix where each element \(J_{ij}\) is given by the partial derivative of the Cartesian coordinate \(x_i\) with respect to the spherical coordinate \(q_j\) (\(x, y, z\) with respect to \(\rho, \theta, \phi\)). So, the matrix \(J\) is:\[ J = \begin{bmatrix} \frac{\partial x}{\partial \rho} & \frac{\partial x}{\partial \theta} & \frac{\partial x}{\partial \phi} \ \frac{\partial y}{\partial \rho} & \frac{\partial y}{\partial \theta} & \frac{\partial y}{\partial \phi} \ \frac{\partial z}{\partial \rho} & \frac{\partial z}{\partial \theta} & \frac{\partial z}{\partial \phi} \end{bmatrix} \].

Step 3: Compute Partial Derivatives

Calculate each element of the Jacobian matrix:\[ \frac{\partial x}{\partial \rho} = \sin \phi \cos \theta \] \[ \frac{\partial x}{\partial \theta} = -\rho \sin \phi \sin \theta \] \[ \frac{\partial x}{\partial \phi} = \rho \cos \phi \cos \theta \]\[ \frac{\partial y}{\partial \rho} = \sin \phi \sin \theta \] \[ \frac{\partial y}{\partial \theta} = \rho \sin \phi \cos \theta \] \[ \frac{\partial y}{\partial \phi} = \rho \cos \phi \sin \theta \] \[ \frac{\partial z}{\partial \rho} = \cos \phi \] \[ \frac{\partial z}{\partial \theta} = 0 \] \[ \frac{\partial z}{\partial \phi} = -\rho \sin \phi \].

Step 4: Assemble the Jacobian Matrix

Now, substitute the computed partial derivatives into the Jacobian matrix:\[ J = \begin{bmatrix} \sin \phi \cos \theta & -\rho \sin \phi \sin \theta & \rho \cos \phi \cos \theta \ \sin \phi \sin \theta & \rho \sin \phi \cos \theta & \rho \cos \phi \sin \theta \ \cos \phi & 0 & -\rho \sin \phi \end{bmatrix} \].

Step 5: Calculate the Determinant of the Jacobian Matrix

The Jacobian determinant, also known as the Jacobian, is obtained by calculating the determinant of the assembled matrix. The determinant for this \(3 \times 3\) matrix is found using the standard formula for determinants. After expanding the determinant, we find:\[ \text{Det}(J) = \rho^2 \sin \phi \].

Step 6: Interpret the Jacobian

The Jacobian \(\rho^2 \sin \phi\) represents the factor by which volumes transform under the change of coordinates from rectangular to spherical. This factor shows how a small region in Cartesian coordinates changes when expressed in spherical coordinates.

Key Concepts

Rectangular to Spherical Transformation

The conversion from rectangular (Cartesian) to spherical coordinates addresses how we describe the position of points in space using different systems. In rectangular coordinates, a point is represented by

• \(x\): the horizontal distance from the origin along the x-axis.
• \(y\): the horizontal distance along the y-axis.
• \(z\): the vertical distance from the origin.

For spherical coordinates, a point is elucidated as

• \(\rho\): the radial distance from the origin.
• \(\theta\): the azimuthal (horizontal) angle from the positive x-axis.
• \(\phi\): the polar (vertical) angle from the positive z-axis.

This transformation involves a set of equations:

• x is converted using \(x = \rho \sin \phi \cos \theta\), representing the horizontal projection.
• y is given by \(y = \rho \sin \phi \sin \theta\), another projection onto the plane.
• z uses \(z = \rho \cos \phi\), indicating the vertical axis alignment.

This translation aids in deeper geometrical understanding and simplifies certain integrals in physics and engineering.

Coordinate Systems

A coordinate system is a powerful tool in mathematics and physics that allows us to describe a point's position in space. Whether dealing with one-dimensional cases like a line, or more complex three-dimensional spaces, coordinate systems are fundamental.
There are several types of coordinate systems, but the two primarily discussed here are:

• **Rectangular (Cartesian) Coordinates**: Use ordered pairs or triplets, such as \((x, y)\) in two dimensions or \((x, y, z)\) in three dimensions, to locate a point based on perpendicular axes.
• **Spherical Coordinates**: Ideal for spherical or radial symmetry problems, particularly useful in three-dimensional space, characterized by a radial distance (\(\rho\)), a polar angle (\(\phi\)), and an azimuthal angle (\(\theta\)).

Each system has its own advantages and fits different kinds of geometrical or physical situations. Understanding them enables us to switch between descriptions of space, thereby allowing more comprehensive solutions to various spatial problems.

Partial Derivatives

Partial derivatives are the building blocks of multivariable calculus. These derivatives are indispensable when dealing with functions of multiple variables as they allow us to examine how changes in one variable affect the function while keeping the others constant.

• For example, if we have a function \(f(x, y)\), the partial derivative \(\frac{\partial f}{\partial x}\) measures how \(f\) changes as \(x\) changes, keeping \(y\) constant.
• Similarly, \(\frac{\partial f}{\partial y}\) reveals how \(f\) changes with respect to \(y\).

In the context of coordinate transformations, partial derivatives are crucial for forming the Jacobian matrix. By calculating the partial derivatives of the coordinate transformation equations (e.g., from spherical to Cartesian), we populate this matrix. This process translates into understanding the sensitivity of each Cartesian coordinate to changes in spherical coordinates. It forms the foundation for more complex concepts like the Jacobian determinant.

Jacobian Determinant

The Jacobian determinant is an essential concept in multivariable calculus, particularly when transforming coordinates. It provides a measure of how much a transformation distorts, scales, or warps space.
For a transformation from rectangular

• to spherical coordinates, the Jacobian determinant is derived by taking the determinant of the Jacobian matrix (whose elements are the partial derivatives of the transformation functions.)
• In our scenario, this gives us \(\text{Det}(J) = \rho^2 \sin \phi\).

This determinant value is not just a number; it physically represents the volume scaling factor. When, for instance, integrating functions over a volume in spherical coordinates, this factor adjusts the integral to account for the coordinate system's curvature and directionality properties.