Express the following sets in rectangular coordinates and identify the set. Assume a is a positive real number. (i) {(p,4,0): p = 2a cos y,0 ≤ y ≤, 0≤ 0 ≤ 2} (ii) { (p, q, 0): p = 4 sec 4,0 ≤ y ≤ 1,0 ≤ 0 ≤ 2}
Express the following sets in rectangular coordinates and identify the set. Assume a is a positive real number. (i) {(p,4,0): p = 2a cos y,0 ≤ y ≤, 0≤ 0 ≤ 2} (ii) { (p, q, 0): p = 4 sec 4,0 ≤ y ≤ 1,0 ≤ 0 ≤ 2}
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Transcription for Educational Website:**
**Title: Conversion of Spherical to Rectangular Coordinates**
**Objective:**
Express the following sets in rectangular coordinates and identify the set. Assume \( a \) is a positive real number.
---
**Problem Statement:**
1. **Set (i):**
\[
\{ ( \rho, \varphi, \theta ): \rho = 2a \cos \varphi, \, 0 \leq \varphi \leq \frac{\pi}{2}, \, 0 \leq \theta \leq 2\pi \}
\]
2. **Set (ii):**
\[
\{ ( \rho, \varphi, \theta ): \rho = 4 \sec \varphi, \, 0 \leq \varphi \leq \frac{\pi}{2}, \, 0 \leq \theta \leq 2\pi \}
\]
**Explanation:**
- **Spherical to Rectangular Coordinate Conversion:**
- \(\rho\) is the radial distance,
- \(\varphi\) is the polar angle,
- \(\theta\) is the azimuthal angle.
The conversion formulas are:
\[
x = \rho \sin \varphi \cos \theta
\]
\[
y = \rho \sin \varphi \sin \theta
\]
\[
z = \rho \cos \varphi
\]
**Graph/Diagram Details:**
- There are no graphs or diagrams provided in the image for these problems.
- The focus is on converting the spherical coordinates to determine the set and identifying their representation in rectangular coordinates.
**Analysis:**
- **Set (i):** This defines a surface within the constraints of \(\rho = 2a \cos \varphi\), serving as a geometric section in 3D space.
- **Set (ii):** Similar to Set (i), this defines a surface using \(\rho = 4 \sec \varphi\).
- By substituting into the coordinate conversions and simplifying, one can identify whether these surfaces represent familiar geometric forms such as planes, cylinders, or spheres.
In your educational journey, analyzing these conversions helps understand the relationship between different coordinate systems and their applications in mathematical modeling and physics](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdc743a43-7060-46cf-ba18-e67544dd4d48%2Fa44eafa3-0792-4a25-9bc2-20d2e0e611d9%2Fcc0oyn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Transcription for Educational Website:**
**Title: Conversion of Spherical to Rectangular Coordinates**
**Objective:**
Express the following sets in rectangular coordinates and identify the set. Assume \( a \) is a positive real number.
---
**Problem Statement:**
1. **Set (i):**
\[
\{ ( \rho, \varphi, \theta ): \rho = 2a \cos \varphi, \, 0 \leq \varphi \leq \frac{\pi}{2}, \, 0 \leq \theta \leq 2\pi \}
\]
2. **Set (ii):**
\[
\{ ( \rho, \varphi, \theta ): \rho = 4 \sec \varphi, \, 0 \leq \varphi \leq \frac{\pi}{2}, \, 0 \leq \theta \leq 2\pi \}
\]
**Explanation:**
- **Spherical to Rectangular Coordinate Conversion:**
- \(\rho\) is the radial distance,
- \(\varphi\) is the polar angle,
- \(\theta\) is the azimuthal angle.
The conversion formulas are:
\[
x = \rho \sin \varphi \cos \theta
\]
\[
y = \rho \sin \varphi \sin \theta
\]
\[
z = \rho \cos \varphi
\]
**Graph/Diagram Details:**
- There are no graphs or diagrams provided in the image for these problems.
- The focus is on converting the spherical coordinates to determine the set and identifying their representation in rectangular coordinates.
**Analysis:**
- **Set (i):** This defines a surface within the constraints of \(\rho = 2a \cos \varphi\), serving as a geometric section in 3D space.
- **Set (ii):** Similar to Set (i), this defines a surface using \(\rho = 4 \sec \varphi\).
- By substituting into the coordinate conversions and simplifying, one can identify whether these surfaces represent familiar geometric forms such as planes, cylinders, or spheres.
In your educational journey, analyzing these conversions helps understand the relationship between different coordinate systems and their applications in mathematical modeling and physics
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