Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![**Problem Statement:**
7. Change the point \((-2, 2, 1)\) from rectangular to cylindrical coordinates and spherical coordinates.
**Solution:**
The given point in rectangular coordinates is \((-2, 2, 1)\).
**For Cylindrical Coordinates:**
- The radial distance \( r \) is calculated using the formula \( r = \sqrt{x^2 + y^2} \).
- The angle \( \theta \) is calculated using \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \).
- The height \( z \) remains the same as in rectangular coordinates.
**For Spherical Coordinates:**
- The radial distance \( \rho \) is calculated using the formula \( \rho = \sqrt{x^2 + y^2 + z^2} \).
- The polar angle \( \phi \) is calculated using \( \phi = \cos^{-1}\left(\frac{z}{\rho}\right) \).
- The azimuthal angle \( \theta \) is the same as in cylindrical coordinates.
The image shows part of the calculation for \(\rho\) as:
\[
\rho = \sqrt{x^2 + y^2 + z^2} = \sqrt{(-2)^2 + 2^2 + 1^2} = 3
\]
This provides the radial distance for the spherical coordinates.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8327d6e3-5c77-497b-a90c-d5ce68ac92e0%2F78b4d812-a5fc-4b27-a9a7-1306dab740e2%2Fa8ya2dd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
7. Change the point \((-2, 2, 1)\) from rectangular to cylindrical coordinates and spherical coordinates.
**Solution:**
The given point in rectangular coordinates is \((-2, 2, 1)\).
**For Cylindrical Coordinates:**
- The radial distance \( r \) is calculated using the formula \( r = \sqrt{x^2 + y^2} \).
- The angle \( \theta \) is calculated using \( \theta = \tan^{-1}\left(\frac{y}{x}\right) \).
- The height \( z \) remains the same as in rectangular coordinates.
**For Spherical Coordinates:**
- The radial distance \( \rho \) is calculated using the formula \( \rho = \sqrt{x^2 + y^2 + z^2} \).
- The polar angle \( \phi \) is calculated using \( \phi = \cos^{-1}\left(\frac{z}{\rho}\right) \).
- The azimuthal angle \( \theta \) is the same as in cylindrical coordinates.
The image shows part of the calculation for \(\rho\) as:
\[
\rho = \sqrt{x^2 + y^2 + z^2} = \sqrt{(-2)^2 + 2^2 + 1^2} = 3
\]
This provides the radial distance for the spherical coordinates.
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