Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 17-30, make the required change in the given equation. \(x^{2}+y^{2}+4 z^{2}=10\) to spherical coordinates

Step-by-Step Solution

Answer

The spherical equation is \(\rho = \sqrt{\frac{10}{1 + 3 \cos^2(\phi)}}\)."

Step 1: Understanding the Problem

We need to convert the equation from Cartesian coordinates \(x, y, z\) to spherical coordinates \(\rho, \theta, \phi\). Spherical coordinates relate to Cartesian coordinates as follows: \(x = \rho \sin(\phi) \cos(\theta)\), \(y = \rho \sin(\phi) \sin(\theta)\), and \(z = \rho \cos(\phi)\).

Step 2: Substitute Spherical Coordinates

Replace \(x, y, z\) in the original equation with their spherical counterparts. The equation becomes \([\rho \sin(\phi) \cos(\theta)]^2 + [\rho \sin(\phi) \sin(\theta)]^2 + 4[\rho \cos(\phi)]^2 = 10\).

Step 3: Simplify the Equation

Simplify the equation by noting \(\sin^2(\theta) + \cos^2(\theta) = 1\). The equation reduces to \(\rho^2 \sin^2(\phi) + 4 \rho^2 \cos^2(\phi) = 10\), which can be rewritten as \(\rho^2 (\sin^2(\phi) + 4 \cos^2(\phi)) = 10\).

Step 4: Factor and Further Simplify

Factor \(\rho^2\) out of the equation. This becomes \(\rho^2 (1 + 3 \cos^2(\phi)) = 10\). Thus, the equation in spherical coordinates is \(\rho^2 (1 + 3 \cos^2(\phi)) = 10\).

Step 5: Solve for \(\rho\)

Solve for \(\rho\) to express it in terms of \(\phi\). \(\rho = \sqrt{\frac{10}{1 + 3 \cos^2(\phi)}}\).

Key Concepts

Coordinate Conversion

Coordinate conversion is the process of changing the representation of a given point or set of points from one coordinate system to another. In our example, we need to convert the equation from Cartesian coordinates \(x, y, z\) to spherical coordinates \(\rho, \theta, \phi\). Changing these coordinates allows us to represent geometric shapes and spatial figures more conveniently.
To convert from Cartesian to spherical coordinates:

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

By substituting these equations into the original equation, we can begin transforming a Cartesian equation to the spherical form.

Cartesian Coordinates

Cartesian coordinates, introduced by René Descartes, are the standard \(x, y, z\) coordinate system used in mathematics. With this system, we can represent any point in three-dimensional space by specifying its horizontal, vertical, and depth position with respect to a fixed origin.
The simplest way to understand Cartesian coordinates is by imagining the three axes:

• The x-axis represents the horizontal dimension.
• The y-axis represents the vertical dimension.
• The z-axis represents depth or height.

In our exercise, the equation \(x^2 + y^2 + 4z^2 = 10\) uses Cartesian coordinates to describe a geometrical relation, which we aim to convert to a spherical form.

Spherical Coordinate System

The spherical coordinate system is a way of expressing points in space using three different values: radial distance, azimuthal angle, and polar angle. These coordinates are essential when dealing with problems involving spheres or circular motions.
Here's what the spherical coordinates mean:

• \(\rho\) - the radial distance from the origin to the point.
• \(\theta\) - the azimuthal angle measured in the x-y plane from the positive x-axis.
• \(\phi\) - the polar angle measured from the positive z-axis.

With spherical coordinates, complex equations representing circles and spheres become more manageable. For the provided exercise, we express Cartesian components \(x, y, z\) in terms of \(\rho, \theta, \phi\) to transform the equation.

Mathematical Problem-Solving

Mathematical problem-solving involves understanding the problem, identifying the goal, and developing a strategy to reach the solution. In coordinate conversion problems, this process can provide clarity and facilitate easier calculations.
Here's a structured approach:

• Understand the equations: Identify the existing form (Cartesian) and the desired form (spherical).
• Substitute variables: Use the known relations between Cartesian and spherical coordinates for substitution.
• Simplify equations: Apply trigonometric identities such as \(\sin^2(\theta)+\cos^2(\theta)=1\) to simplify expressions.
• Solve for key variables: Reduce the resultant form to express necessary variables explicitly, like \(\rho\) in terms of \(\phi\) in our solution.

Using these steps can ensure a methodical approach to tackling similar problems, allowing for coherent and accurate solutions.