Problem 59 - Advanced Engineering Mathematics

Question

Convert the point given in spherical coordinates to (a) rectangular coordinates and (b) cylindrical coordinates. $$ \left(\frac{2}{3}, \frac{\pi}{2}, \frac{\pi}{6}\right) $$

Step-by-Step Solution

Answer

Rectangular: (0, 1/3, \(\sqrt{3}/3\)); Cylindrical: (1/3, \(\pi/2\), \(\sqrt{3}/3\)).

Step 1: Understanding Spherical Coordinates

The given spherical coordinates are: \( r = \frac{2}{3} \), \( \theta = \frac{\pi}{2} \), and \( \phi = \frac{\pi}{6} \). In this system, \( r \) is the radial distance from the origin, \( \theta \) is the azimuthal angle in the xy-plane from the positive x-axis, and \( \phi \) is the polar angle from the positive z-axis.

Step 2: Convert to Rectangular Coordinates

To convert from spherical to rectangular coordinates, use the formulas: \( x = r \sin(\phi) \cos(\theta) \), \( y = r \sin(\phi) \sin(\theta) \), and \( z = r \cos(\phi) \). Substitute the values:\[ x = \left(\frac{2}{3}\right) \sin\left(\frac{\pi}{6}\right) \cos\left(\frac{\pi}{2}\right) = \left(\frac{2}{3}\right) \cdot \frac{1}{2} \cdot 0 = 0 \]\[ y = \left(\frac{2}{3}\right) \sin\left(\frac{\pi}{6}\right) \sin\left(\frac{\pi}{2}\right) = \left(\frac{2}{3}\right) \cdot \frac{1}{2} \cdot 1 = \frac{1}{3} \]\[ z = \left(\frac{2}{3}\right) \cos\left(\frac{\pi}{6}\right) = \left(\frac{2}{3}\right) \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{3} \]

Step 3: Rectangular Coordinates Output

Thus, the rectangular coordinates of the point are \( (0, \frac{1}{3}, \frac{\sqrt{3}}{3}) \).

Step 4: Convert to Cylindrical Coordinates

To convert from spherical to cylindrical coordinates, use \( \rho = r \sin(\phi) \), \( \theta = \theta \), and \( z = r \cos(\phi) \). Substitute the values:\[ \rho = \left(\frac{2}{3}\right) \sin\left(\frac{\pi}{6}\right) = \left(\frac{2}{3}\right) \cdot \frac{1}{2} = \frac{1}{3} \]\[ \theta = \frac{\pi}{2} \]\[ z = \left(\frac{2}{3}\right) \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{3} \]

Step 5: Cylindrical Coordinates Output

Thus, the cylindrical coordinates of the point are \( \left(\frac{1}{3}, \frac{\pi}{2}, \frac{\sqrt{3}}{3}\right) \).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe a point in space using three axes mutually perpendicular to each other: x, y, and z. Each point is defined by how far along it is on each of these axes, offering a straightforward and intuitive way to pinpoint locations in three-dimensional space.

In the context of coordinate conversion, transforming a point from spherical to rectangular coordinates involves using trigonometric functions. The formulas are:

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

In this method, \( r \) is the radial distance from the origin, \( \theta \) is the azimuthal angle, and \( \phi \) is the polar angle. Coordinates such as \( (0, \frac{1}{3}, \frac{\sqrt{3}}{3}) \) tell us that the point is on the y-axis and above the x-y plane. This makes it easy to visualize locations in fields such as physics and engineering.

Cylindrical Coordinates

Cylindrical coordinates provide another way of representing points in space. They are particularly useful in problems involving symmetry around an axis, commonly seen in engineering and physical systems. In this system, a point is described with three parameters: \( \rho \), \( \theta \), and \( z \).

• \( \rho \) represents the radial distance from the vertical z-axis, similar to the radius in polar coordinates but extended into three dimensions.
• \( \theta \) is the azimuthal angle, representing the direction around the z-axis.
• \( z \) remains the same as in rectangular coordinates, reflecting height above or below the x-y plane.

To convert from spherical to cylindrical coordinates, use the formulas:

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

The specified point \( \left(\frac{1}{3}, \frac{\pi}{2}, \frac{\sqrt{3}}{3}\right) \) indicates a position that is radially \( \frac{1}{3} \) units from the z-axis and has a height from the x-y plane defined by \( \frac{\sqrt{3}}{3} \).

Coordinate Conversion

Coordinate conversion is a crucial skill in mathematics, physics, and engineering. It allows one to translate a point's description from one coordinate system to another, highlighting the point's geometry relative to different axes or focal points.

When dealing with spherical to rectangular or cylindrical conversions, understanding the trigonometric identities involved is key. The calculations often incorporate:

• Sine and cosine functions that manage angles relative to rotational and axial symmetry.
• Formulas that adjust for the change from radial distances to cartesian positions.

In practice, knowing how to efficiently convert coordinates can simplify complex problems, making it easier to analyze and compute results from multiple perspectives. Whether dealing with spheres, tubes, or complex geometries, this skill ensures clarity and precision in spatial reasoning and problem-solving.