Problem 45 - Advanced Engineering Mathematics

Question

In Problems \(43-46\), convert the given equation to cylindrical coardinates. $$ x^{2}+y^{2}-z^{2}=1 $$

Step-by-Step Solution

Answer

The equation in cylindrical coordinates is \(r^2 - z^2 = 1\).

Step 1: Understanding the Problem

The problem requires converting the equation from Cartesian coordinates \((x, y, z)\) to cylindrical coordinates \((r, \theta, z)\). In cylindrical coordinates, the relations are \(x = r\cos(\theta)\), \(y = r\sin(\theta)\), and \(z = z\).

Step 2: Converting Terms

We observe that \(x^2 + y^2\) is equivalent to \(r^2\) in cylindrical coordinates. Thus, replace \(x^2 + y^2\) with \(r^2\).

Step 3: Substitute into the Equation

The original equation is \(x^2 + y^2 - z^2 = 1\). Substitute \(r^2\) for \(x^2 + y^2\) to get: \(r^2 - z^2 = 1\).

Step 4: Verify the Substitution

Confirm that each term is correctly rewritten in cylindrical coordinates. The term \(x^2 + y^2\) correctly translates to \(r^2\), and the \(z\)-term remains unchanged as \(z^2\).

Key Concepts

Cartesian Coordinates

Cartesian coordinates provide a way to describe the position of a point in three-dimensional space using three values: \(x\), \(y\), and \(z\). These values represent positions along the horizontal (x-axis), vertical (y-axis), and the depth (z-axis), respectively. In this coordinate system, the position of any point can be related to its distance and direction from a fixed point called the origin. The Cartesian system is often visualized as a grid of perpendicular lines, making it easy to specify any location with a set of numerical values.
One of the primary reasons Cartesian coordinates are so widely used is their straightforward application in various fields such as physics, engineering, and computer graphics. They provide a simple and intuitive way to solve geometric problems. When given an equation like \(x^2 + y^2 - z^2 = 1\), it's initially expressed in Cartesian terms, describing a surface in three-dimensional space.
Understanding this system allows for an easier grasp on how transformations like coordinate conversion can be seamlessly performed.

Coordinate Systems

Coordinate systems are methods used to uniquely identify the position of a point in space. Various systems exist to accommodate different dimensions and applications, such as Cartesian, cylindrical, spherical, and others. Each has its unique way of expressing points based on the application's requirements or the problem's nature.
The cylindrical coordinate system, as mentioned in the problem, is especially useful when dealing with circular or helical structures. It focuses on the radial distance from a central axis (denoted as \(r\)), the angle around the axis (called \(\theta\)), and the height along the axis (notated as \(z\)). This system simplifies problems involving symmetry around a line, as in the case of cylinders or circular patterns.
When switching between coordinate systems, understanding the properties of each system aids in making accurate translations. For instance, the transformation from Cartesian to cylindrical coordinates hinges on understanding the relations like \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\), which help to map one system to another seamlessly.

Coordinate Conversion

Coordinate conversion is the process of changing from one coordinate system to another. This conversion is essential when the situation or problem geometry aligns more naturally with a different coordinate system. It simplifies equations, making them easier to solve or understand. In the exercise, converting from Cartesian to cylindrical coordinates made handling circular symmetry much more intuitive.
To perform such conversions, it’s crucial to know the relationships between the systems involved. For Cartesian to cylindrical conversion, these are given by:

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

Applying these relations, we replaced \(x^2 + y^2\) with \(r^2\), reflecting it as the radial distance squared in cylindrical terms. Following through the problem helped confirm that the translation maintained the same characteristics as the original equation: \(r^2 - z^2 = 1\).
Understanding coordinate conversions is fundamental, especially where problems showcase symmetries or properties better suited to a different system. It allows for more elegant solutions and greater insight into the potential geometry of the problem.