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Chapter 14: Problem 1

Explain how cylindrical coordinates are used to describe a point in \(\mathbb{R}^{3}\).

Short Answer

Expert verified
Answer: To find the cylindrical coordinates of a point in three-dimensional space, follow these steps: 1. Define cylindrical coordinates as a representation of a point in space using \((\rho, \theta, z)\). 2. Understand the conversion equations between Cartesian and cylindrical coordinates. 3. Find the radial distance \(\rho\) by calculating the square root of the sum of the squared \(x\) and \(y\) components. 4. Find the azimuthal angle \(\theta\) using the inverse tangent function of the ratio of the \(y\) and \(x\) components. 5. Determine the height \(z\), which is the same as the \(z\)-component in Cartesian coordinates.

Step by step solution

01

Define cylindrical coordinates

Cylindrical coordinates are a three-dimensional coordinate system used to describe the position of a point in space. They are especially useful for problems with cylindrical symmetry. A point in cylindrical coordinates is represented as \((\rho, \theta, z)\), where \(\rho\) is the radial distance from the \(z\)-axis, \(\theta\) is the azimuthal angle measured from the positive \(x\)-axis, and \(z\) is the height above the \(xy\)-plane.
02

Conversion between Cartesian and cylindrical coordinates

To convert between the Cartesian coordinates \((x, y, z)\) and the cylindrical coordinates \((\rho, \theta, z)\) of a point, we use the following equations: \begin{align*} x &= \rho \cos{\theta}, \\ y &= \rho \sin{\theta}, \\ z &= z. \end{align*} Conversely, we can find the cylindrical coordinates using these equations: \begin{align*} \rho &= \sqrt{x^2 + y^2}, \\ \theta &= \tan^{-1}\frac{y}{x}, \\ z &= z. \end{align*}
03

Find the radial distance \(\rho\)

Given a point with Cartesian coordinates \((x, y, z)\), find the radial distance \(\rho\) by calculating the square root of the sum of the squared \(x\) and \(y\) components: $$ \rho = \sqrt{x^2 + y^2}. $$
04

Find the azimuthal angle \(\theta\)

To find the azimuthal angle \(\theta\), use the inverse tangent function (arctangent) of the ratio of the \(y\) and \(x\) components: $$ \theta = \tan^{-1}\frac{y}{x}. $$ Note that if \(x = 0\), \(\theta\) will be undefined. In this case, check the value of \(y\) to determine the angle. If \(y > 0\), \(\theta = \frac{\pi}{2}\). If \(y < 0\), \(\theta = \frac{3\pi}{2}\).
05

Find the height \(z\)

In cylindrical coordinates, the height \(z\) of a point is the same as its \(z\)-component in Cartesian coordinates: $$ z = z. $$ By following these steps, we can describe any point in \(\mathbb{R}^3\) using spherical coordinates.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conversion between Cartesian and Cylindrical Coordinates
Understanding how to switch between Cartesian and cylindrical coordinates is essential in many fields, such as physics and engineering. These conversions allow us to choose the most convenient system for solving problems.

To convert from Cartesian
  • **Cartesian to Cylindrical:** - \( x = \rho \cos{\theta} \) - \( y = \rho \sin{\theta} \) - \( z = z \)
  • **Cylindrical to Cartesian:** - \( \rho = \sqrt{x^2 + y^2} \) - \( \theta = \tan^{-1}\frac{y}{x} \) - \( z = z \)
This bidirectional conversion makes it easier to work in whichever coordinate system is more suited to the situation, offering clarity and precision.
Radial Distance
The radial distance \( \rho \) is a key part of cylindrical coordinates. It measures how far a point is from the vertical \( z \)-axis. Think of it almost like the radius of a circle projected in the \( xy \)-plane.

The formula to find the radial distance from Cartesian coordinates
  • \( \rho = \sqrt{x^2 + y^2} \)
This formula makes it simple to calculate \( \rho \) using the familiar constructs of distance in the plane.
Azimuthal Angle
The azimuthal angle \( \theta \) is essential for locating a point's direction in the \( xy \)-plane. It is similar to how a compass or protractor measures angles from a standard direction, here the positive \( x \)-axis.

To find \( \theta \):
  • Use the formula: \( \theta = \tan^{-1}\frac{y}{x} \)
  • Consider special cases: If \( x = 0 \): - With \( y > 0 \), \( \theta = \frac{\pi}{2} \) - With \( y < 0 \), \( \theta = \frac{3\pi}{2} \)
The azimuthal angle provides a clear understanding of directionality, enhancing spatial awareness.
Coordinate Systems
Coordinate systems, like the Cartesian and cylindrical systems, are frameworks that help us describe positions in space. Each has its special applications and advantages.

Cylindrical coordinates are particularly useful when dealing with objects that have rotational symmetry, like cylinders and tubes. They allow for a more intuitive description of points in such contexts, often simplifying the mathematics involved.

In contrast, Cartesian coordinates are best for problems that involve straightforward grid-based layouts.
  • **Cartesian:** Straight-line measurements along axes
  • **Cylindrical:** Focuses on distance from a central axis and angle
Choosing the right coordinate system can make solving problems more intuitive and efficient.

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Most popular questions from this chapter

Let \(R\) be the region bounded by the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1,\) where \(a > 0\) and \(b>0\) are real numbers. Let \(T\) be the transformation \(x=a u, y=b v\). Find the center of mass of the upper half of \(R(y \geq 0)\) assuming it has a constant density.

A thin plate of unit density occupies the region between the parabola \(y=a x^{2}\) and the horizontal line \(y=b\) where \(a > 0\) and \(b > 0 .\) Show that the center of mass is \(\left(0, \frac{3 b}{5}\right),\) independent of \(a\)

Use a change of variables to evaluate the following integrals. \(\iiint_{D} d V ; D\) is bounded by the upper half of the ellipsoid \(x^{2} / 9+y^{2} / 4+z^{2}=1\) and the \(x y\) -plane. Use \(x=3 u\) \(y=2 v, z=w\)

Evaluate the following integrals. $$\iint_{R} y d A ; R=\\{(x, y): 0 \leq y \leq \sec x, 0 \leq x \leq \pi / 3\\}$$

Use the definition for the average value of a function over a region \(R\) (Section 1 ), \(\bar{f}=\frac{1}{\text { area of } R} \iint_{R} f(x, y) d A\). Find the average value of \(a-x-y\) over the region \(R=\\{(x, y): x+y \leq a, x \geq 0, y \geq 0\\},\) where \(a>0\)
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