Problem 25 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 17-30, make the required change in the given equation. \(x+y=4\) to cylindrical coordinates

Step-by-Step Solution

Answer

The equation in cylindrical coordinates is \(r = \frac{4}{\cos\theta + \sin\theta}\).

Step 1: Understand Cylindrical Coordinates

In cylindrical coordinates, we describe points in three dimensions using \((r, \theta, z)\). The relationship between cylindrical and Cartesian coordinates \((x, y, z)\) is given by: 1. \(x = r \cos \theta\) 2. \(y = r \sin \theta\) 3. \(z = z\) For the presented problem, the equation is in the xy-plane (no z component), so we will primarily convert the \(x\) and \(y\) components.

Step 2: Substitute Coordinates

Substitute the cylindrical coordinate expressions for \(x\) and \(y\) into the equation \(x + y = 4\): \[ r\cos\theta + r\sin\theta = 4 \]

Step 3: Factor Out the Common Term

Factor out the common factor \(r\) from the terms on the left-hand side: \[ r(\cos\theta + \sin\theta) = 4 \]

Step 4: Solve for r

To express \(r\) in terms of \(\theta\), divide both sides by the expression \((\cos\theta + \sin\theta)\): \[ r = \frac{4}{\cos\theta + \sin\theta} \]

Key Concepts

Coordinate Transformation

Coordinate transformation is an essential technique in mathematics and engineering, allowing us to shift between different systems of expressing coordinates. It involves converting information from one system into another, making complex problems easier to solve.

• In three-dimensional space, coordinates can be described in various forms, including Cartesian, cylindrical, or spherical.
• Each type of coordinate system has its own advantages depending on the symmetry and nature of the problem being analyzed.

For instance, cylindrical coordinates are often beneficial when dealing with problems involving cylinders or circular symmetry. By effectively using transformations, it’s easier to navigate complex geometry and solve intricate mathematical problems using the right approach.

Cartesian Coordinates

Cartesian coordinates are commonly used in mathematics and everyday applications to denote the position of a point in space.

• In two dimensions, Cartesian coordinates are represented as \(x, y\) points in a plane.
• In three dimensions, they expand to \(x, y, z\), giving a complete description of a location in space.

The Cartesian system is very intuitive since it divides space into perpendicular axes, making measurement straightforward. This simplicity makes it a foundational element in mathematics education, helping learners visualize geometric and algebraic concepts clearly.
When transforming to cylindrical coordinates, we derive the relation of these rectangular components to cylindrical parameters like \(r\) and \(\theta\). Knowing these conversions is vital for conducting various calculations more efficiently in different fields of study.

Mathematics Education

Mathematics education focuses on developing critical thinking, problem-solving skills, and an understanding of abstract concepts through a variety of mathematical disciplines. Teaching coordinate transformations are a part of advanced mathematics education topics.

• Students learn various coordinate systems to enhance their spatial reasoning ability.
• Working with different coordinate systems assists students in visualizing and solving problems in physics, engineering, and other sciences.

Understanding how to convert between coordinate systems broadens students' ability to interpret and tackle real-world problems. These skills are not only crucial in academia but also in numerous professional fields, providing a basis for further study and applications.

Equation Conversion

Equation conversion is the process of rewriting equations in a different form to facilitate solving or interpretation of a problem. In this exercise, converting the equation \(x + y = 4\) into cylindrical coordinates simplifies the system by aligning the equation with the nature of cylindrical symmetry.

• This process involves substituting the Cartesian coordinates \(x\) and \(y\) with expressions \(r \cos \theta\) and \(r \sin \theta\) that represent these components in cylindrical terms.
• From here, manipulation such as factoring and solving for \(r\) allows the equation to be fully expressed in terms of cylindrical coordinates.

Understanding and mastering such conversions are valuable skills as they make solving certain equations easier and better suited to automation, simulation, and analysis in technology and engineering contexts.