Problem 61 - Advanced Engineering Mathematics

Question

In Problems \(59-62\), convert the point given in spherical coordinates to (a) rectangular coordinates and (b) cylindrical coordinates. $$ \left(8, \frac{\pi}{4}, \frac{3 \pi}{4}\right) $$

Step-by-Step Solution

Answer

Rectangular: (4, 4, -4); Cylindrical: (4√2, π/4, -4).

Step 1: Definitions and Formulas

To convert a point from spherical coordinates \(( ho, \theta, \phi)\) to rectangular coordinates \((x, y, z)\), we use:\[x = \rho \sin\phi \cos\theta\]\[y = \rho \sin\phi \sin\theta\]\[z = \rho \cos\phi\]

Step 2: Apply Rectangular Coordinate Formulas

Given the point \((\rho, \theta, \phi) = (8, \frac{\pi}{4}, \frac{3\pi}{4})\):- Calculate \(x\) using \[x = 8 \sin\frac{3\pi}{4} \cos\frac{\pi}{4}\].- Calculate \(y\) using \[y = 8 \sin\frac{3\pi}{4} \sin\frac{\pi}{4}\].- Calculate \(z\) using \[z = 8 \cos\frac{3\pi}{4}\].

Step 3: Compute x, y, z

First, find the sine and cosine values:- \(\sin\frac{3\pi}{4} = \frac{\sqrt{2}}{2}\)- \(\cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}\)- \(\sin\frac{\pi}{4} = \frac{\sqrt{2}}{2}\)- \(\cos\frac{3\pi}{4} = -\frac{\sqrt{2}}{2}\)Now calculate:- \(x = 8 \times \frac{\sqrt{2}}{2} \times \frac{\sqrt{2}}{2} = 4 \)- \(y = 8 \times \frac{\sqrt{2}}{2} \times \frac{\sqrt{2}}{2} = 4\)- \(z = 8 \times -\frac{\sqrt{2}}{2} = -4\)So, the point in rectangular coordinates is \((4, 4, -4)\).

Step 4: Definitions for Cylindrical Coordinates

To convert from spherical coordinates \((\rho, \theta, \phi)\) to cylindrical coordinates \((r, \theta, z)\), use the following formulas:\[r = \rho \sin\phi\]\[\theta = \theta\]\[z = \rho \cos\phi\]

Step 5: Apply Cylindrical Coordinate Formulas

Given the same point \((\rho, \theta, \phi) = (8, \frac{\pi}{4}, \frac{3\pi}{4})\):- Calculate \(r\) using \[r = 8 \sin\frac{3\pi}{4}\].- Keep \(\theta = \frac{\pi}{4}\).- Calculate \(z = 8 \cos\frac{3\pi}{4}\).

Step 6: Compute r, \theta, z

Use previously found values:- \(\sin\frac{3\pi}{4} = \frac{\sqrt{2}}{2}\)- \(\cos\frac{3\pi}{4} = -\frac{\sqrt{2}}{2}\)Compute:- \(r = 8 \times \frac{\sqrt{2}}{2} = 4\sqrt{2}\)- \(\theta = \frac{\pi}{4}\)- \(z = 8 \times -\frac{\sqrt{2}}{2} = -4\)So, the point in cylindrical coordinates is \((4\sqrt{2}, \frac{\pi}{4}, -4)\).

Key Concepts

Spherical Coordinates

Spherical coordinates offer a unique way to describe a point in three-dimensional space. They consist of three components:

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

These components help locate a point by assessing its position on a sphere, providing a useful perspective especially in problems involving spheres and circular symmetry.
For example, the point given in the exercise \(8, \frac{\pi}{4}, \frac{3\pi}{4}\) indicates a distance of 8 units from the origin, an angle of 45° or \(\frac{\pi}{4}\) radians along the xy-plane, and a 135° or \(\frac{3\pi}{4}\) radians downward from the z-axis.
This system is particularly useful in physics, as many natural phenomena exhibit spherical symmetry, simplifying calculations and concepts.

Rectangular Coordinates

Rectangular, or Cartesian coordinates, provide a familiar way to represent points in space, using standard \(x, y, z\) format.
These coordinates are straightforward as they denote the orthogonal projections onto the x, y, and z axes respectively.

• \(x\): Project onto the x-axis.
• \(y\): Project onto the y-axis.
• \(z\): Project onto the z-axis.

In our conversion from spherical coordinates (example given is \(8, \frac{\pi}{4}, \frac{3\pi}{4}\)), we use the following formulas:

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

For this specific exercise, calculating these gives the rectangular coordinates \(4, 4, -4\).
They are suitable for straightforward distance and vector operations, playing a crucial role in mechanics and engineering.

Cylindrical Coordinates

Cylindrical coordinates combine elements of both spherical and rectangular systems, particularly useful for problems involving circular symmetry.
This system uses:

• \(r\): The radial distance from the z-axis (similar to the radius in a circle).
• \(\theta\): The angle around the z-axis, like the azimuthal angle in polar coordinates.
• \(z\): The height above the xy-plane, similar to the rectangular coordinate.

To convert from spherical, we adjust the formulas slightly:
For the example point \(8, \frac{\pi}{4}, \frac{3\pi}{4}\), the conversion to cylindrical coordinates would involve:

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

This results in cylindrical coordinates \(4\sqrt{2}, \frac{\pi}{4}, -4\).
Cylindrical coordinates are ideal in applications involving cylinders, like in electromagnetic fields and fluid dynamics.

Trigonometric Functions

Trigonometric functions such as sine and cosine are essential in performing coordinate conversions. They help in determining the components of vectors based on angles.
In conversions like spherical to rectangular or cylindrical, the following is important:

• \(\sin\phi\): Determines vertical alignment with respect to the z-axis.
• \(\cos\phi\): Helps calculate the projection length along the z-axis.
• \(\sin\theta\) and \(\cos\theta\): Resolve the direction in the xy-plane.

For example, in the conversion process detailed above, we utilized:

• \(\sin\frac{3\pi}{4} = \frac{\sqrt{2}}{2}\)
• \(\cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}\)

This facilitated the computation of the final rectangular and cylindrical coordinates.
Understanding these functions enriches the ability to manipulate and translate points between different coordinate systems.