Problem 6 - Calculus Early Transcendentals: Pearson New International Edition

Question

Change the following from Cartesian to cylindrical coordinates. (a) \((2,2,3)\) (b) \((4 \sqrt{3},-4,6)\)

Step-by-Step Solution

Answer

(a) (2√2, π/4, 3); (b) (8, -π/6, 6).

Step 1: Understand Cartesian to Cylindrical Conversion

To convert from Cartesian coordinates \(x, y, z\) to cylindrical coordinates \(r, \theta, z\), we use the equations \(r = \sqrt{x^2 + y^2}\), \(\theta = \arctan\left(\frac{y}{x}\right)\), and the z-coordinate remains the same.

Step 2: Convert (a) (2,2,3)

For the point \(2, 2, 3\):- Calculate \(r\): \( r = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2}\).- Calculate \(\theta\): \( \theta = \arctan\left(\frac{2}{2}\right) = \arctan(1) = \frac{\pi}{4} ext{ radians}\).- The z-coordinate is unchanged, so \(z = 3\).The cylindrical coordinates are \(\left( 2\sqrt{2}, \frac{\pi}{4}, 3 \right)\).

Step 3: Convert (b) (4√3,-4,6)

For the point \(4\sqrt{3}, -4, 6\):- Calculate \(r\): \( r = \sqrt{(4\sqrt{3})^2 + (-4)^2} = \sqrt{48 + 16} = \sqrt{64} = 8\).- Calculate \(\theta\): \( \theta = \arctan\left(\frac{-4}{4\sqrt{3}}\right) = \arctan\left(-\frac{1}{\sqrt{3}}\right) = -\frac{\pi}{6} ext{ radians}\).- The z-coordinate is unchanged, so \(z = 6\).The cylindrical coordinates are \( (8, -\frac{\pi}{6}, 6)\).

Key Concepts

Cartesian Coordinates

The Cartesian coordinate system is a fundamental concept in mathematics, representing points in a two-dimensional plane with pairs of numerical values.
These values are known as the x-coordinate and the y-coordinate. This system can also be extended to three dimensions by adding a z-coordinate.
The idea is to provide a unique location for any point based on its distance from two or three perpendicular axes.
Some key characteristics of Cartesian coordinates include:

• Simplicity: Easy to understand due to its straightforward grid-based design.
• Orthogonal Axes: The x, y, and z axes are perpendicular to each other.
• Support for Algebraic Operations: Facilitates calculations and graphical representations through simple algebraic expressions.

Cartesian coordinates are the foundational building block for many mathematical concepts and are used extensively throughout mathematics education.

Coordinate Conversion

Converting between different coordinate systems is an important skill in mathematics.
It allows us to interpret and solve problems more effectively using the system best suited for the task.
In this case, we are converting from Cartesian to cylindrical coordinates.
The conversion involves the following steps:

• Calculate the radial distance: This is done using the formula \[ r = \sqrt{x^2 + y^2} \]
• Find the angle, \(\theta\): Determine the direction by using the trigonometric function \[ \theta = \arctan \left(\frac{y}{x}\right) \]
• Retain the z-coordinate: This value remains unchanged during the conversion.

By applying these steps, we gain a new perspective on the original Cartesian coordinates, often simplifying problem-solving in contexts like physics and engineering.

Mathematics Education

Mathematics education strives to equip students with the skills necessary to tackle a wide array of problems.
Learning about coordinate systems is a crucial component of this educational journey.
Concepts like coordinate conversions demonstrate how mathematics is more than just numbers and equations - it's about understanding patterns and relationships.
Educational benefits of learning coordinate systems include:

• Visualization: Helps students visualize and interpret multi-dimensional problems.
• Problem-solving: Enhances their capacity to convert theoretical problems into applied mathematics.
• Real-world Applications: Prepares students for careers in fields such as engineering, physics, and computer science.

Emphasizing these skills in mathematics education aids in developing a stronger foundation for students, empowering them to apply mathematical concepts in practical, real-world scenarios.

Trigonometric Functions

Trigonometric functions are pivotal in converting Cartesian coordinates to cylindrical coordinates.
They provide the necessary tools to calculate angles and distances, which are integral to understanding spatial relationships.
Functions like \(\sin\), \(\cos\), and \(\tan\) serve as the building blocks for these calculations.
Some important aspects of trigonometric functions include:

• Angle Calculation: \( \theta = \arctan \left(\frac{y}{x}\right) \) helps determine the direction of the point relative to the x-axis.
• Relationship with Circles: Trigonometric functions link the geometry of circles with algebraic functions, explaining circular and oscillatory motions.
• Application in Various Fields: Used in physics, engineering, and computer science to solve various spatial problems.

Grasping these functions is essential for students to proficiently handle tasks involving different coordinate systems and to deepen their understanding of mathematical and physical phenomena.