Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 17-30, make the required change in the given equation. \(x^{2}+y^{2}=9\) to cylindrical coordinates

Step-by-Step Solution

Answer

The equation in cylindrical coordinates is \(r^{2} = 9\).

Step 1: Substitute Cylindrical Coordinates Formula

To convert the given equation from Cartesian coordinates to cylindrical coordinates, recall that the relationships between these coordinate systems are as follows: \(x = r\cos\theta\) and \(y = r\sin\theta\).

Step 2: Replace x and y in the Equation

Substitute the cylindrical coordinates into the given equation: \(x^{2} + y^{2} = 9\) becomes \((r\cos\theta)^{2} + (r\sin\theta)^{2} = 9\).

Step 3: Simplify the Equation

Simplify the expression \((r\cos\theta)^{2} + (r\sin\theta)^{2}\) using the Pythagorean identity: \(\cos^{2}\theta + \sin^{2}\theta = 1\). So, \(r^{2}(\cos^{2}\theta + \sin^{2}\theta) = r^{2} = 9\).

Step 4: Final Cylindrical Coordinate Equation

The simplified equation in cylindrical coordinates is \(r^{2} = 9\).

Key Concepts

Coordinate Systems

Coordinate systems are essential tools in mathematics and physics that allow us to describe the location of points in space. Each system provides a unique method to specify these points. The most common systems you'll encounter include Cartesian, cylindrical, and spherical coordinates.

• Cartesian Coordinates: Represented by the coordinates \((x, y, z)\) in three-dimensional space, where each axis is perpendicular to the others.
• Cylindrical Coordinates: Utilize \((r, \theta, z)\) to specify locations, where \(r\) is the radial distance from a reference axis, \(\theta\) is the angular coordinate, and \(z\) is the height along the axis.
• Spherical Coordinates: These are represented as \((\rho, \phi, \theta)\), but we'll focus on cylindrical here.

Understanding how to convert between these systems is crucial for solving many problems. Each system is useful in different contexts, such as using cylindrical coordinates for problems involving rotational symmetry.

Pythagorean Identity

The Pythagorean identity is a widely used equation in trigonometry that forms the backbone of many conversions in coordinate systems. This identity is expressed as \(\cos^{2}\theta + \sin^{2}\theta = 1\). This relationship is derived from the Pythagorean theorem, which applies to right-angled triangles.

• Key Uses: The identity helps in simplifying equations that involve squared trigonometric functions, making calculations easier.
• Application: In our exercise, the identity simplifies the equation \((r\cos\theta)^{2} + (r\sin\theta)^{2}\), directly transforming it into \(r^{2}\).

By replacing \((r\cos\theta)^{2} + (r\sin\theta)^{2}\) with \(r^{2}\) using the identity, you derive the simpler equation \(r^{2} = 9\), showing how powerful this identity can be for mathematical conversions.

Cartesian Coordinates

Cartesian coordinates are the foundation of many mathematical and engineering calculations, especially when dealing with linear and planar geometry. In a Cartesian coordinate system, each point in the space is defined by \((x, y, z)\) coordinates in three dimensions, or \((x, y)\) in two dimensions. Each axis represents a spatial dimension and is perpendicular to the others.

• Origin: The point \((0,0)\) or \((0,0,0)\) is known as the origin, serving as the reference point for all other points.
• Advantages: This system is particularly useful for plotting equations and visualizing geometric shapes.
• Conversion: When converting a Cartesian equation to cylindrical coordinates, such as in our exercise, you replace \(x\) with \(r\cos\theta\) and \(y\) with \(r\sin\theta\), as shown in the step-by-step solution.

Grasping the Cartesian system deeply informs your ability to switch seamlessly among various coordinate systems, adapting easily to the unique demands of each mathematical problem.