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 determinant is \( \rho^2 \sin\phi \).

Step 1: Understand the Coordinate Transformation

In spherical coordinates, the variables \( (x, y, z) \) in rectangular coordinates are transformed to \( (\rho, \theta, \phi) \), where \( \rho \) is the radius, \( \theta \) is the azimuthal angle, and \( \phi \) is the polar angle. The relationships are given as: \( x = \rho \sin\phi \cos\theta \), \( y = \rho \sin\phi \sin\theta \), and \( z = \rho \cos\phi \).

Step 2: Set Up the Jacobian Matrix

The Jacobian matrix for the transformation is a 3x3 matrix of partial derivatives. We will compute the partial derivatives of \( x, y, z \) with respect to \( \rho, \theta, \phi \).\[J = \begin{bmatrix}\frac{\partial x}{\partial \rho} & \frac{\partial x}{\partial \theta} & \frac{\partial x}{\partial \phi} \\frac{\partial y}{\partial \rho} & \frac{\partial y}{\partial \theta} & \frac{\partial y}{\partial \phi} \\frac{\partial z}{\partial \rho} & \frac{\partial z}{\partial \theta} & \frac{\partial z}{\partial \phi}\end{bmatrix}\]

Step 3: Compute the Partial Derivatives

Calculate the partial derivatives for each component:\( \frac{\partial x}{\partial \rho} = \sin\phi \cos\theta \),\( \frac{\partial x}{\partial \theta} = -\rho \sin\phi \sin\theta \),\( \frac{\partial x}{\partial \phi} = \rho \cos\phi \cos\theta \).\( \frac{\partial y}{\partial \rho} = \sin\phi \sin\theta \),\( \frac{\partial y}{\partial \theta} = \rho \sin\phi \cos\theta \),\( \frac{\partial y}{\partial \phi} = \rho \cos\phi \sin\theta \).\( \frac{\partial z}{\partial \rho} = \cos\phi \),\( \frac{\partial z}{\partial \theta} = 0 \),\( \frac{\partial z}{\partial \phi} = -\rho \sin\phi \).

Step 4: Formulate the Jacobian Matrix

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

The Jacobian determinant needed for the transformation is calculated as:\[\det(J) = \rho^2 \sin\phi\]This determinant represents the factor by which volume changes when transforming from rectangular to spherical coordinates.

Key Concepts

Spherical Coordinates

Spherical coordinates are a system of three variables used to describe the location of a point in three-dimensional space. They are denoted by \( (\rho, \theta, \phi) \):

• \( \rho \): The radial distance from the origin to the point. It represents how far the point is from the origin, like a radius.
• \( \theta \): Known as the azimuthal angle, \( \theta \) is measured in the \( x-y \) plane from the positive \( x \)-axis. It tells us the direction of the point around a circle.
• \( \phi \): The polar angle measured from the positive \( z \)-axis down towards the \( x-y \) plane. This angle shows the elevation of the point.

Spherical coordinates are particularly useful when dealing with symmetrical objects like spheres, as they simplify the mathematics involved.

Rectangular Coordinates

Rectangular coordinates, also called Cartesian coordinates, are the most straightforward method to pinpoint a location in space. They use three variables: \( (x, y, z) \), each corresponding to axes perpendicular to each other:

• \( x \): Specifies the horizontal position along the x-axis.
• \( y \): Indicates the position along the y-axis, which is perpendicular to the x-axis.
• \( z \): Denotes the vertical position along the z-axis, standing perpendicular to both the x and y axes.

Together, these three numbers create a coordinate that describes a point's specific location within a three-dimensional space. Rectangular coordinates are especially convenient for calculations involving straight lines and right angles.

Coordinate Transformation

A coordinate transformation involves converting a set of coordinates from one system to another. When moving from rectangular to spherical coordinates, this transformation follows specific equations:

• \( x = \rho \sin\phi \cos\theta \)
• \( y = \rho \sin\phi \sin\theta \)
• \( z = \rho \cos\phi \)

These equations allow us to translate a location in three-dimensional rectangular coordinates to spherical coordinates, reflecting aspects like radial distance, azimuthal, and polar angles. Coordinate transformations are crucial in fields that require different views of data, such as engineering and physics, enabling more manageable calculations in contexts like radial symmetry.

Partial Derivatives

Partial derivatives are a fundamental concept in calculus used to measure how a function changes as each variable changes independently. In the context of coordinate transformations, partial derivatives are instrumental:

• They provide the rates of change for one set of coordinates with respect to another.
• In the Jacobian matrix, each element represents a partial derivative, showing how each rectangular coordinate \( (x, y, z) \) transforms into spherical coordinates \( (\rho, \theta, \phi) \).
• These derivatives help compute the Jacobian determinant, which relates to how volume transforms during a change of coordinates.

Using partial derivatives, we can calculate components like

• \( \frac{\partial x}{\partial \rho} = \sin\phi \cos\theta \)
• \( \frac{\partial x}{\partial \theta} = -\rho \sin\phi \sin\theta \)
• \( \frac{\partial x}{\partial \phi} = \rho \cos\phi \cos\theta \)

These calculations are vital for understanding how transformations affect the geometry of objects in three-dimensional space.