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2. Change the point (2,4,-4) from rectangular to cylindrical coordinates and spherical coordinates.

2. Change the point (2,4,-4) from rectangular to cylindrical coordinates and spherical coordinates.

Calculus: Early Transcendentals
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:** 2. Change the point (2, 4, -4) from rectangular to cylindrical coordinates and spherical coordinates. --- **Explanation:** This problem requires converting the given point in rectangular (Cartesian) coordinates `(x, y, z)` to both cylindrical and spherical coordinate systems. **Cylindrical Coordinates:** The cylindrical coordinate system `(r, θ, z)` is related to Cartesian coordinates `(x, y, z)` using the following relationships: - \( r = \sqrt{x^2 + y^2} \) - \( θ = \tan^{-1}(\frac{y}{x}) \) - \( z = z \) **Spherical Coordinates:** The spherical coordinate system `(ρ, θ, φ)` is related to Cartesian coordinates `(x, y, z)` using the following relationships: - \( ρ = \sqrt{x^2 + y^2 + z^2} \) - \( θ = \tan^{-1}(\frac{y}{x}) \) - \( φ = \cos^{-1}(\frac{z}{ρ}) \) **Steps to solve:** 1. **Cylindrical Coordinates:** - Calculate \( r \) using \( r = \sqrt{2^2 + 4^2} = \sqrt{20} = 2\sqrt{5} \). - Calculate \( θ \) using \( θ = \tan^{-1}(\frac{4}{2}) = \tan^{-1}(2) \). - The \( z \)-coordinate remains the same, i.e., \( z = -4 \). 2. **Spherical Coordinates:** - Calculate \( ρ \) using \( ρ = \sqrt{2^2 + 4^2 + (-4)^2} = \sqrt{36} = 6 \). - The angle \( θ \) is the same as in cylindrical coordinates, \( θ = \tan^{-1}(2) \). - Calculate \( φ \) using \( φ = \cos^{-1}(\frac{-4}{6}) = \cos^{-1}(-\frac{2}{3}) \). Substitute these values back into the respective coordinate forms to obtain the cylindrical and spherical coordinates.
Transcribed Image Text:**Problem Statement:** 2. Change the point (2, 4, -4) from rectangular to cylindrical coordinates and spherical coordinates. --- **Explanation:** This problem requires converting the given point in rectangular (Cartesian) coordinates `(x, y, z)` to both cylindrical and spherical coordinate systems. **Cylindrical Coordinates:** The cylindrical coordinate system `(r, θ, z)` is related to Cartesian coordinates `(x, y, z)` using the following relationships: - \( r = \sqrt{x^2 + y^2} \) - \( θ = \tan^{-1}(\frac{y}{x}) \) - \( z = z \) **Spherical Coordinates:** The spherical coordinate system `(ρ, θ, φ)` is related to Cartesian coordinates `(x, y, z)` using the following relationships: - \( ρ = \sqrt{x^2 + y^2 + z^2} \) - \( θ = \tan^{-1}(\frac{y}{x}) \) - \( φ = \cos^{-1}(\frac{z}{ρ}) \) **Steps to solve:** 1. **Cylindrical Coordinates:** - Calculate \( r \) using \( r = \sqrt{2^2 + 4^2} = \sqrt{20} = 2\sqrt{5} \). - Calculate \( θ \) using \( θ = \tan^{-1}(\frac{4}{2}) = \tan^{-1}(2) \). - The \( z \)-coordinate remains the same, i.e., \( z = -4 \). 2. **Spherical Coordinates:** - Calculate \( ρ \) using \( ρ = \sqrt{2^2 + 4^2 + (-4)^2} = \sqrt{36} = 6 \). - The angle \( θ \) is the same as in cylindrical coordinates, \( θ = \tan^{-1}(2) \). - Calculate \( φ \) using \( φ = \cos^{-1}(\frac{-4}{6}) = \cos^{-1}(-\frac{2}{3}) \). Substitute these values back into the respective coordinate forms to obtain the cylindrical and spherical coordinates.
Expert Solution
Step 1: Introduction

Given: The rectangular coordinate points 2,4,-4

Find the cylindrical and spherical coordinates from the given point.

The cylindrical coordinates is of the form r,θ,z

The spherical coordinates is of the form ρ,ϕ,θ

 

 

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