Problem 64 - Advanced Engineering Mathematics

Question

In Problems \(63-66\), convert the points given in rectangular coordinates to spherical coordinates. $$ (1,-\sqrt{3}, 1) $$

Step-by-Step Solution

Answer

The spherical coordinates are \(\left(\sqrt{5}, \frac{2\pi}{3}, \cos^{-1}\left(\frac{1}{\sqrt{5}}\right)\right)\).

Step 1: Understanding the Relationship

Spherical coordinates (\(\rho, \theta, \phi\)) are related to rectangular coordinates (\(x, y, z\)) by the following equations:\[\rho = \sqrt{x^2 + y^2 + z^2}\]\[\theta = \tan^{-1}(y/x)\]\[\phi = \cos^{-1}(z/\rho)\].

Step 2: Calculate \(\rho\)

Find the value of \(\rho\) using the formula:\[\rho = \sqrt{x^2 + y^2 + z^2}\]Substitute the given points (\(1, -\sqrt{3}, 1\)) into the equation:\[\rho = \sqrt{1^2 + (-\sqrt{3})^2 + 1^2} = \sqrt{1 + 3 + 1} = \sqrt{5}\].

Step 3: Calculate \(\theta\)

\(\theta\) is calculated as:\[\theta = \tan^{-1}\left(\frac{y}{x}\right)\]Substitute \(y = -\sqrt{3}\) and \(x = 1\):\[\theta = \tan^{-1}\left(\frac{-\sqrt{3}}{1}\right) = \tan^{-1}(-\sqrt{3})\]\(\tan^{-1}(-\sqrt{3})\) is \(-\frac{\pi}{3}\).However, to fit spherical coordinates (range \(0\) to \(2\pi\)), add \(\pi\):\[\theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3}\].

Step 4: Calculate \(\phi\)

\(\phi\) is calculated as:\[\phi = \cos^{-1}\left(\frac{z}{\rho}\right)\]Substitute \(z = 1\) and \(\rho = \sqrt{5}\):\[\phi = \cos^{-1}\left(\frac{1}{\sqrt{5}}\right)\].

Step 5: Express in Spherical Coordinates

The spherical coordinates are \((\rho, \theta, \phi)\) where \(\rho = \sqrt{5}\), \(\theta = \frac{2\pi}{3}\), and \(\phi = \cos^{-1}\left(\frac{1}{\sqrt{5}}\right)\). Thus, the spherical coordinates are \(\left(\sqrt{5}, \frac{2\pi}{3}, \cos^{-1}\left(\frac{1}{\sqrt{5}}\right)\right)\).

Key Concepts

Coordinate Systems

A coordinate system is a systematic way of assigning a set of numbers, or coordinates, to each point in a space, so that any point can be uniquely identified. These systems provide a framework for positions in space using numerical references. Among many types, the most common are rectangular (also called Cartesian) and spherical coordinate systems.
Rectangular coordinates use axes that intersect at right angles, where a position is described by

• The x-coordinate, the horizontal distance from the origin.
• The y-coordinate, the vertical distance from the origin.
• The z-coordinate, height above or below a reference plane.

Spherical coordinates, on the other hand, use distance from an origin and angles to define a position in space:

• \( \rho \): The radial distance from the origin.
• \( \theta \): The azimuthal angle in the xy-plane from the x-axis.
• \( \phi \): The polar angle from the z-axis.

These coordinate systems allow for versatility and adaptability in different mathematical and scientific applications.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, form the foundation of many fields in science and engineering. By using three orthogonal axes—x, y, and z—a point's location in three-dimensional space is uniquely defined.
This system is straightforward because:

• It divides space into sectors, where each value refers to a specific sector.
• Every point may be expressed as \( (x, y, z) \).

Consider the example point \((1, -\sqrt{3}, 1)\). Each number corresponds directly to its position along these axes. Thus, understanding this system is crucial in navigating through topics involving space geometry and algebraic calculations.
The intuitive nature of rectangular coordinates makes them excellent for simple tasks like plotting graphs or solving basic geometry problems. But, for more complex shapes or orientations, converting to a curvilinear system like spherical coordinates often simplifies matters.

Mathematical Conversions

Mathematical conversions allow us to transform a point from one coordinate system to another. In the example given, converting from rectangular to spherical coordinates provides a different perspective on the geometrical placement in space.
Conversion involves using mathematical equations to transform coordinates. Here's how we accomplished it for point \((1, -\sqrt{3}, 1)\):

• \( \rho \): Calculated as \( \rho = \sqrt{x^2 + y^2 + z^2} \). By substituting in the values, we find \( \rho = \sqrt{5} \).
• \( \theta \): Initially \( \theta = \tan^{-1} (y/x) \). Adjust it to fit within the standard range for spherical coordinates, resulting in \( \theta = \frac{2\pi}{3} \).
• \( \phi \): Given by \( \phi = \cos^{-1}(z/\rho) \), which provides \( \phi = \cos^{-1}\left(\frac{1}{\sqrt{5}}\right) \).

Conversions between coordinate systems are powerful tools in solving complex problems. They allow us to choose the most convenient perspective for analysis. These techniques, while sometimes computationally intensive, uncover the symmetry and simplicity that might be obscured in a different system.