Problem 18 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 17-30, make the required change in the given equation. \(x^{2}-y^{2}=25\) to cylindrical coordinates

Step-by-Step Solution

Answer

The equation in cylindrical coordinates is \( r^{2} \cos(2\theta) = 25 \).

Step 1: Understand Cylindrical Coordinates

Cylindrical coordinates are a 3D coordinate system that extend polar coordinates with a height value. In cylindrical coordinates, \( (x, y, z) \) is represented as \( (r, \theta, z) \). The relations are: \( x = r \cos(\theta) \), \( y = r \sin(\theta) \), and \( z = z \).

Step 2: Substitute for x and y

Substitute \( x = r \cos(\theta) \) and \( y = r \sin(\theta) \) into the equation \( x^{2} - y^{2} = 25 \). So it becomes: \( (r \cos(\theta))^{2} - (r \sin(\theta))^{2} = 25 \).

Step 3: Simplify the Expression

The expression \( (r \cos(\theta))^{2} - (r \sin(\theta))^{2} \) simplifies to:\( r^{2} \cos^{2}(\theta) - r^{2} \sin^{2}(\theta) \).

Step 4: Factor Out Common Terms

Factor out \( r^{2} \) from the terms:\( r^{2} (\cos^{2}(\theta) - \sin^{2}(\theta)) = 25 \).

Step 5: Apply Double Angle Identity

Apply the trigonometric identity \( \cos^{2}(\theta) - \sin^{2}(\theta) = \cos(2\theta) \):\( r^{2} \cos(2\theta) = 25 \).

Step 6: Rewrite the Equation

The equation in cylindrical coordinates is \( r^{2} \cos(2\theta) = 25 \). This expresses the original equation in terms of \( r \) and \( \theta \).

Key Concepts

Polar Coordinates

Polar coordinates are essential in understanding how to transition between different types of coordinate systems. Instead of using rectangular coordinates with an x-axis and y-axis, polar coordinates represent points in a plane using a distance and an angle. Here, a point is defined by

• r: the radial distance from the origin.
• θ: the angle measured from the positive x-axis.

For example, the coordinate (2, π/4) in polar coordinates indicates a point that is 2 units away from the origin and 45 degrees (or π/4 radians) counterclockwise from the positive x-axis. This framework is particularly useful in situations with circular symmetry or where distance and direction are more relevant than horizontal and vertical placement.
Using polar coordinates can simplify math problems by transforming complex equations into simpler forms, especially when symmetry or circular motion is involved.

Trigonometric Identities

Trigonometric identities are equations that are true for all angles and are useful in simplifying expressions. In this exercise, the identity \( ext{cos}^{2}(θ) - ext{sin}^{2}(θ) = ext{cos}(2θ)\) is used. This specific identity allows us to reduce the expression involving trigonometric functions during the coordinate conversion process.
Here’s why this identity is handy:

• The formula can be used to simplify expressions from a squared sine and cosine form into a single cosine term.
• It directly connects squared trigonometric terms with a single angle expression, easing calculations in polar or cylindrical transformations.

A thorough understanding of such identities not only helps in this conversion exercise but also broadens your problem-solving toolkit in calculus and other areas of mathematics.

Coordinate Transformation

Coordinate transformation is the process of changing from one coordinate system to another. In this exercise, we shift from Cartesian to cylindrical coordinates using formulas that relate the two systems. Cylindrical coordinates are an extension of polar coordinates into three dimensions.
For a point in the 2D plane, we use:

• \(x = r \cos(θ)\)
• \(y = r \sin(θ)\)

Stepping into three dimensions, cylindrical coordinates add a vertical component, denoted as \(z\). So, the transformation includes maintaining the \(z\) value while changing \(x\) and \(y\) to polar forms.
This process simplifies dealing with rotations and symmetries, especially in problems involving geometry and physics, making them much easier to solve.

Calculus

Calculus often involves transforming coordinates to make derivatives and integrations simpler. When working with cylindrical coordinates, calculus problems often use the modified volume and area elements suitable for these systems.
In calculus, whether dealing with differentiation or integration, utilizing the right coordinate system can transform a challenging problem into an easier one:

• Integrals over circular regions become straightforward.
• Symmetric problems, especially in physical contexts like gravitational fields or electromagnetism, can be handled more systematically.

Through coordinate transformation and the use of trigonometric identities, you employ calculus to explore deeper properties of functions and their relationships to real-world phenomena.