Problem 72 - Advanced Engineering Mathematics

Question

In Problems \(71-74\), convert the given equation to rectangular coordinates. $$ \phi=\pi / 3 $$

Step-by-Step Solution

Answer

The equation in rectangular coordinates represents a plane defined by \( z = \frac{1}{2}r \).

Step 1: Understand Polar Coordinates

The equation \( \phi = \pi/3 \) is given in polar coordinates, where \( \phi \) represents the angle made with the positive \( z \)-axis in spherical coordinates. In this context, however, it seems we need to treat \( \phi \) as the angular component in polar coordinates, typically related to \( \theta \) in planar polar coordinates, describing a 2D position.

Step 2: Define Conversion Relationships

In polar coordinates, the conversion between polar and rectangular coordinates involves \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \). However, in a 3D spherical context, \( z = r \cos(\phi) \) could also be considered.

Step 3: Interpret the Given Equation

For our problem, it seems we're looking to convert \( \phi = \pi/3 \) under the assumption of its relation to traditional polar angle, meaning this translates directly to determining which specific plane or section of space is being described in rectangular coordinates.

Step 4: Express the Plane in Rectangular Coordinates

In the context of the problem, \( \phi = \pi/3 \) relates to cylindrical or spherical geometry, signifying a surface where every point on it has the angle \( \phi = \pi/3 \). This corresponds to the conic plane in spherical coordinates where \( z = r \cos(\pi/3) \). Since \( \cos(\pi/3) = 1/2 \), the equation in rectangular coordinates can be expressed as \( z = \frac{1}{2}r \), where \( r = \sqrt{x^2 + y^2 + z^2} \).

Key Concepts

Polar Coordinates

Polar coordinates offer a unique way to describe a point in a plane using a radius and an angle, rather than using x and y coordinates as in rectangular form. In this system, a point is represented by the coordinate pair \(r, \theta\).
\(\begin{align*} r & : \text{the distance from the origin to the point}\ \theta & : \text{the angle between the positive x-axis and the line connecting the origin with the point}\end{align*}\)
This approach is particularly advantageous in scenarios involving circular or rotational symmetry.

• Simplicity: Polar coordinates simplify the representation and calculation of problems involving curves like spirals and circles.
• Conversion: To switch from polar to rectangular coordinates, use:\(\begin{align*} x &= r \cos(\theta) \ y &= r \sin(\theta)\end{align*}\)
• Visual Nature: Visualizing problems becomes easier as many natural symmetries align with a polar grid.

Understanding polar coordinates can make tasks involving rotations and angles more intuitive and manageable.

Spherical Coordinates

Spherical coordinates extend the concept of polar coordinates into three dimensions, allowing us to specify locations on a sphere's surface. In this system, a point in space is defined by three coordinates: \(r, \, \theta, \, \phi\).

• Radius \(r\): It's the distance from the origin to the point.
• Azimuthal Angle \(\theta\): It measures the angle in the xy-plane from the positive x-axis.
• Zenith Angle \(\phi\): This is the angle from the positive z-axis down to the radius.

The convenience of spherical coordinates lies in their ability to naturally describe systems with radial symmetry, like planets or stars.
In relation to conversion to rectangular coordinates:
\(\begin{align*} x &= r \sin(\phi) \cos(\theta) \ y &= r \sin(\phi) \sin(\theta) \ z &= r \cos(\phi)\end{align*}\)
These coordinates make calculating gravitational fields, electromagnetic fields, and modeling other 3D phenomena much clearer.

Conversion Equations

Conversion equations are crucial as they enable the translation between different coordinate systems, such as polar, spherical, and rectangular. This allows us to choose the most efficient system for a particular problem.

• Polar to Rectangular: Useful for 2D rotations and simplifications:\(\begin{align*} x &= r \cos(\theta) \ y &= r \sin(\theta)\end{align*}\)
• Spherical to Rectangular: Ideal for 3D situations where symmetry around a point is prominent:\(\begin{align*} x &= r \sin(\phi) \cos(\theta) \ y &= r \sin(\phi) \sin(\theta) \ z &= r \cos(\phi)\end{align*}\)
• Rectangular to Polar/Spherical: Convert back when needed for analysis or specific calculations.

Each system has its strengths. Polar and spherical coordinates excel in simplifying complex circular or spherical problems, while rectangular coordinates are versatile for general use. Mastering conversion between these systems equips you to tackle a broad range of mathematical and physical challenges.