Problem 46 - Advanced Engineering Mathematics

Question

Conven the given equation to cylindrical codrdinates. $$ x^{2}+z^{2}=16 $$

Step-by-Step Solution

Answer

The equation in cylindrical coordinates is \(r^2 + z^2 = 16\).

Step 1: Identify the Cartesian Coordinates

The given equation is in Cartesian coordinates: \(x^{2} + z^{2} = 16\), where \(x\), \(y\), and \(z\) are the coordinates in the Cartesian system.

Step 2: Understand the Cylindrical Coordinate System

In cylindrical coordinates, a point is represented by \((r, \theta, z)\) where \(r\) is the radial distance from the origin to the point's projection on the \(xy\)-plane, \(\theta\) is the angle from the positive x-axis, and \(z\) remains the same as in Cartesian coordinates.

Step 3: Substitute the Relation for \(x\) in Cylindrical Coordinates

In cylindrical coordinates, \(x = r\cos(\theta)\). Substitute this into the equation:\[ r^{2}\cos^{2}(\theta) + z^{2} = 16 \]

Step 4: Simplify the Expression Using General Identity for \(r\)

Recall from cylindrical coordinates, another relation is \(r^{2} = x^{2} + y^{2}\). For the equation \(x^{2} + z^{2} = 16\) and \(y=0\), \(r = x\). Therefore, \(x^{2}=r^{2}\).

Step 5: Final Equation in Cylindrical Coordinates

Since \(r^{2} = x^{2}\), the equation \(x^{2} + z^{2} = 16\) becomes \(r^{2} + z^{2} = 16\).This is the final conversion in cylindrical coordinates.

Key Concepts

Cartesian coordinates

The Cartesian coordinate system is one of the most fundamental systems used to locate a point in space. It's composed of axes which, in three-dimensional space, are labeled as the x, y, and z axes. This system is named after René Descartes, and it's widely used due to its simplicity and ease of understanding.
In the Cartesian system:

• x represents the horizontal distance from the origin along the x-axis.
• y represents the horizontal distance from the origin along the y-axis.
• z represents the vertical distance from the origin, or elevation.

All points in this system can be represented as coordinates, denoted as (x, y, z). This system is frequently used because it allows for straightforward calculations involving distances and angles, making it essential in mathematics and physics.

coordinate conversion

Coordinate conversion is a crucial mathematical process that involves transforming coordinates from one system to another. It's essential in various fields such as science, engineering, and navigation, to ensure consistency and accuracy when analyzing data from different sources.
When converting coordinates:

• It's important to understand the characteristics of both coordinate systems involved.
• Formulas and relationships are used to translate coordinates. These depend on the specific systems being converted.

For instance, when converting Cartesian coordinates to cylindrical coordinates, we use specific equations like \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \), taking advantage of the relationship between radial distance, angle, and height.
Conversion helps to maintain the integrity and meaning of geometric data across various scenario applications.

cylindrical coordinate system

The cylindrical coordinate system is an extension of the polar coordinate system into three dimensions. It's particularly useful for simplifying problems involving symmetrical properties about an axis, especially around the z-axis in three-dimensional space.
In cylindrical coordinates, each point in space is depicted as (r, θ, z) where:

• r is the radial distance from the center or origin to the location's projection on the xy-plane.
• θ is the angle measured from the positive x-axis.
• z specifies the elevation, similar to the z-coordinate in Cartesian coordinates.

This system is particularly helpful in fields such as electromagnetism and fluid dynamics, where circular symmetry simplifies complex integrals and equations. Transitioning an equation like \( x^2 + z^2 = 16 \) from Cartesian to cylindrical coordinates involves expressing x in terms of r and θ, simplifying several steps.

radial distance

Radial distance is a concept in coordinate systems that indicates how far a point is from a specific reference point - usually the origin. In the context of cylindrical (and polar) coordinates, the radial distance, r, plays a significant role.
Key points about radial distance:

• In cylindrical coordinates, r is calculated from the Pythagorean theorem as \( r = \sqrt{x^2 + y^2} \).
• It simplifies calculations, especially in scenarios with rotational symmetry.
• Radial distance is critical in converting coordinates, as it effectively "replaces" the traditional x and y Cartesian coordinates in equations.

In our example, the equation \( x^2 + z^2 = 16 \) simplifies to \( r^2 + z^2 = 16 \) in cylindrical coordinates, assuming y = 0, making x equivalent to r. This highlights how radial distance helps in transforming and solving equations efficiently.