Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

Make the required change in the given equation. \(x^{2}+y^{2}+4 z^{2}=10\) to spherical coordinates

Step-by-Step Solution

Answer

In spherical coordinates, \(\rho^2 = \frac{10}{\sin^2\phi + 4\cos^2\phi}\).

Step 1: Understand the Task

We need to express the equation \(x^2 + y^2 + 4z^2 = 10\) in terms of spherical coordinates. In spherical coordinates, point (x, y, z) are expressed as \((\rho, \theta, \phi)\).

Step 2: Know the Spherical Coordinate Transformations

The spherical coordinates are defined as follows: \(x = \rho \sin\phi \cos\theta\), \(y = \rho \sin\phi \sin\theta\), and \(z = \rho \cos\phi\), where \(\theta\) is the azimuthal angle, \(\phi\) is the polar angle, and \(\rho\) is the radial distance from the origin.

Step 3: Substitute Spherical Coordinates into the Equation

Replace \(x\), \(y\), and \(z\) in the equation \(x^2 + y^2 + 4z^2 = 10\) with the spherical coordinate expressions: \((\rho \sin\phi \cos\theta)^2 + (\rho \sin\phi \sin\theta)^2 + 4(\rho \cos\phi)^2 = 10\).

Step 4: Simplify the Equation

Simplifying, we have: \((\rho^2 \sin^2\phi \cos^2\theta) + (\rho^2 \sin^2\phi \sin^2\theta) + 4\rho^2 \cos^2\phi = 10\). Notice that \(\cos^2\theta + \sin^2\theta = 1\), so the equation becomes \(\rho^2 \sin^2\phi + 4\rho^2 \cos^2\phi = 10\).

Step 5: Factor Out Common Terms

Factor out \(\rho^2\) from the equation: \(\rho^2 (\sin^2\phi + 4\cos^2\phi) = 10\).

Step 6: Solve for \(\rho^2\)

The equation \(\rho^2 (\sin^2\phi + 4\cos^2\phi) = 10\) implies that \(\rho^2 = \frac{10}{\sin^2\phi + 4\cos^2\phi}\).

Key Concepts

Coordinate Transformation

In mathematics and physics, coordinate transformation is the process of converting a set of coordinates from one system to another. Different coordinate systems can simplify representation of complex shapes and equations.

For the exercise we're tackling, we aim to transform the rectangular coordinates

• x
• y
• z

into spherical coordinates:

• \( \rho \)
• \( \theta \)
• \( \phi \)

Spherical coordinates are beneficial in scenarios where symmetry exists around a point, such as solving equations involving spheres and circles.

The corresponding transformations are governed by these equations:

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

These transformations allow us to express a three-dimensional point \(x, y, z\) in terms of spherical coordinates \( (\rho, \theta, \phi) \). This change often simplifies complex geometric shapes and enables easier manipulation of equations.

Equation Simplification

Equation simplification in mathematics involves minimizing an expression to its simplest form without changing its value. Converting our original rectangular equation into spherical coordinates was just the start. The tricky part involves simplifying the new spherical equation.

Here's what we had: \[ (\rho \sin\phi \cos\theta)^2 + (\rho \sin\phi \sin\theta)^2 + 4(\rho \cos\phi)^2 = 10 \]
Notice how parts of the equation can simplify using trigonometric identities, such as \[\cos^2\theta + \sin^2\theta = 1\].

Applying these identities, we can first condense the equation to:\[\rho^2 \sin^2\phi + 4\rho^2 \cos^2\phi = 10\]
Further simplification is facilitated by factoring out common terms. Here, we extract \( \rho^2 \), yielding: \[\rho^2 ( \sin^2\phi + 4\cos^2\phi ) = 10\] This form highlights the simplified relationship between the spherical coordinate components and the original equation. Simplifying complex equations aids in the understanding and solving of problems.

Polar Angle

The polar angle, denoted by \( \phi \), is one of the key components in spherical coordinates. It represents the angle between the positive z-axis and the line segment connecting the origin to the point in question.

This angle ranges from 0 to \( \pi \) radians, or equivalently, from 0° to 180°. In practical terms, it measures how "high" or "low" the point is located relative to the plane marked by the x and y axes.

For instance, when \( \phi = 0 \), the point lies on the positive z-axis. Conversely, when \( \phi = \pi/2 \), the point lies in the xy-plane.

• \( \phi = 0 \) : Point is on the positive z-axis.
• \( \phi = \pi/2 \) : Point lies in the xy-plane.
• \( \phi = \pi \) : Point is on the negative z-axis.

The concept of the polar angle is crucial in distinguishing spherical coordinates from other systems, providing a precise measure of a point's z-component. It helps relate the spherical system back to our familiar Cartesian coordinates.

Azimuthal Angle

The azimuthal angle, labeled as \( \theta \), is another essential element of spherical coordinates. This angle takes its measurement from the positive x-axis, usually in the xy-plane, and revolves around the z-axis.

An azimuthal angle ranges from 0 to \( 2\pi \) radians or 0° to 360°. This wide range allows it to capture all possible directional variations in the plane. Here's how it rotates:

• \( \theta = 0 \) : Directed along the positive x-axis.
• \( \theta = \pi/2 \) : Directed along the positive y-axis.
• \( \theta = \pi \) : Directed along the negative x-axis.
• \( \theta = 3\pi/2 \) : Directed along the negative y-axis.

Understanding the azimuthal angle is fundamental for mapping the location of a point around the z-axis, making it invaluable in various physical applications, such as in navigation and astronomy. By considering both the azimuthal and polar angles, we can fully describe any point's position in three-dimensional space.