Problem 60 - Advanced Engineering Mathematics

Question

In Problems \(59-62\), convert the point given in spherical coordinates to (a) rectangular coordinates and (b) cylindrical coordinates. $$ \left(5, \frac{5 \pi}{4}, \frac{2 \pi}{3}\right) $$

Step-by-Step Solution

Answer

Rectangular: \\(\left(-\frac{5 \sqrt{6}}{4}, -\frac{5 \sqrt{6}}{4}, -\frac{5}{2}\right)\\), Cylindrical: \\(\left( \frac{5\sqrt{3}}{2}, \frac{5\pi}{4}, -\frac{5}{2} \right)\\).

Step 1: Understanding Spherical Coordinates

Spherical coordinates are given in the form \(\rho, \theta, \phi\), where \(\rho\) is the radial distance, \(\theta\) is the azimuthal angle measured from the positive x-axis in the x-y plane, and \(\phi\) is the polar angle measured from the positive z-axis.

Step 2: Converting to Rectangular Coordinates (x, y, z)

To convert from spherical to rectangular coordinates, use the formulas: 1. \(x = \rho \sin \phi \cos \theta\) 2. \(y = \rho \sin \phi \sin \theta\) 3. \(z = \rho \cos \phi\).Given \(\rho = 5, \theta = \frac{5\pi}{4}, \phi = \frac{2\pi}{3}\), calculate: \[ x = 5 \times \sin\left(\frac{2\pi}{3}\right) \times \cos\left(\frac{5\pi}{4}\right) = 5 \times \frac{\sqrt{3}}{2} \times \left(-\frac{\sqrt{2}}{2}\right) = -\frac{5\sqrt{6}}{4}. \] \[ y = 5 \times \sin\left(\frac{2\pi}{3}\right) \times \sin\left(\frac{5\pi}{4}\right) = 5 \times \frac{\sqrt{3}}{2} \times \left(-\frac{\sqrt{2}}{2}\right) = -\frac{5\sqrt{6}}{4}. \] \[ z = 5 \times \cos\left(\frac{2\pi}{3}\right) = 5 \times \left(-\frac{1}{2}\right) = -\frac{5}{2}. \] Hence, the rectangular coordinates are \(\left(-\frac{5 \sqrt{6}}{4}, -\frac{5 \sqrt{6}}{4}, -\frac{5}{2}\right)\).

Step 3: Converting to Cylindrical Coordinates (r, θ, z)

Cylindrical coordinates are given in the form \(r, \theta, z\). To convert from spherical to cylindrical coordinates, use: 1. \(r = \rho \sin \phi\) 2. \(\theta\) (remains the same as in spherical)3. \(z = \rho \cos \phi\).Given \(\rho = 5, \theta = \frac{5\pi}{4}, \phi = \frac{2\pi}{3}\), calculate: \[ r = 5 \times \sin\left(\frac{2\pi}{3}\right) = 5 \times \frac{\sqrt{3}}{2} = \frac{5\sqrt{3}}{2}. \] \[ \theta = \frac{5\pi}{4}. \] \[ z = 5 \times \cos\left(\frac{2\pi}{3}\right) = -\frac{5}{2}. \] Thus, the cylindrical coordinates are \(\left( \frac{5\sqrt{3}}{2}, \frac{5\pi}{4}, -\frac{5}{2} \right)\).

Key Concepts

Spherical Coordinates

Spherical coordinates are an efficient way to represent points in three-dimensional space using angles and a distance. The system uses three parameters: \( \rho \), \( \theta \), and \( \phi \).
\( \rho \) is the radial distance from the origin to the point. This is a non-negative value.
\( \theta \) is the azimuthal angle, typically measured in the \( xy \)-plane, from the positive x-axis. The angle can range from 0 to \( 2\pi \) radians.
\( \phi \) is the polar angle, measured from the positive z-axis. This angle can range from 0 to \( \pi \) radians. Here's a quick way to visualize it: imagine you are on the surface of a sphere (in a 3D space). \( \rho \) will tell you how big the sphere is (the distance you have to the center), \( \theta \) describes in which direction around the z-axis you are looking, and \( \phi \) tells you how high or low you are tilting your head from the positive z-direction. Converting from spherical coordinates allows for easier integrations or evaluations on specific surfaces.

Rectangular Coordinates

Rectangular coordinates are the standard "x, y, z" format most commonly encountered in three-dimensional math problems. They form the foundation of algebraic geometry and analysis with a straightforward grid-like representation. Points in this system are plotted by moving along the x-axis, the y-axis, and the z-axis. This makes it intuitive for calculating distances and performing algebraic operations. When converting from spherical to rectangular coordinates, you utilize the following transformations:

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


• \( y = \rho \sin \phi \sin \theta \)


• \( z = \rho \cos \phi \)

This set of equations allows you to determine each coordinate independently within a rectangular space, effectively translating angular and radial data into a Cartesian grid.

Cylindrical Coordinates

Cylindrical coordinates are a hybrid between polar and rectangular coordinate systems. They are particularly useful for problems involving symmetry around an axis, such as a pipe or a can. This system also uses three parameters: \( r \), \( \theta \), and \( z \).

• \( r \) is the radial distance from the z-axis to the projection of the point in the \( xy \)-plane.
• \( \theta \) is the angular position in the \( xy \)-plane, identical to the azimuthal angle in spherical coordinates.
• \( z \) is the height above the z-plane, maintaining its vertical distance as in rectangular coordinates.

To convert from spherical to cylindrical coordinates, use the following steps:

• \( r = \rho \sin \phi \)
• \( \theta = \theta \)
• \( z = \rho \cos \phi \)

Cylindrical coordinates become particularly handy in physical applications like fluid dynamics, electromagnetism, and engineering where angular rotation about an axis is common.