Find 2= equation for the paraboit ż=== 3- (x²+y³) & in cylindrical coording equation[ an
Find 2= equation for the paraboit ż=== 3- (x²+y³) & in cylindrical coording equation[ an
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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Can anyone please help me with this question please? I am stuck on it please help me!
![**Task:** Find an equation for the paraboloid \( z = 3 - (x^2 + y^2) \) in cylindrical coordinates.
**Step-by-Step Explanation:**
1. **Understand the Cartesian Equation:**
The given equation is \( z = 3 - (x^2 + y^2) \).
2. **Convert to Cylindrical Coordinates:**
In cylindrical coordinates, \( x = r \cos \theta \) and \( y = r \sin \theta \), where \( r \) is the radial distance, and \( \theta \) is the angular coordinate. Also, \( x^2 + y^2 = r^2 \).
3. **Substitute in the Equation:**
Replace \( x^2 + y^2 \) with \( r^2 \):
\[
z = 3 - r^2
\]
4. **Final Expression:**
The equation of the paraboloid in cylindrical coordinates is \( z = 3 - r^2 \).
**Note:** The paper contains a placeholder box labeled "equation" where the final cylindrical coordinate equation would be placed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F99d15f92-0bff-4b4d-a47e-2ac33d144271%2Fde9fa5ef-9031-4c4b-a018-ac23cdf40281%2Fahh4i8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Task:** Find an equation for the paraboloid \( z = 3 - (x^2 + y^2) \) in cylindrical coordinates.
**Step-by-Step Explanation:**
1. **Understand the Cartesian Equation:**
The given equation is \( z = 3 - (x^2 + y^2) \).
2. **Convert to Cylindrical Coordinates:**
In cylindrical coordinates, \( x = r \cos \theta \) and \( y = r \sin \theta \), where \( r \) is the radial distance, and \( \theta \) is the angular coordinate. Also, \( x^2 + y^2 = r^2 \).
3. **Substitute in the Equation:**
Replace \( x^2 + y^2 \) with \( r^2 \):
\[
z = 3 - r^2
\]
4. **Final Expression:**
The equation of the paraboloid in cylindrical coordinates is \( z = 3 - r^2 \).
**Note:** The paper contains a placeholder box labeled "equation" where the final cylindrical coordinate equation would be placed.
![**Problem:**
Find an equation for the paraboloid \( z = 3 - (x^2 + y^2) \) in cylindrical coordinates.
**Solution:**
To express the given equation in cylindrical coordinates, recall that in cylindrical coordinates, the relationships are:
- \( x = r \cos \theta \)
- \( y = r \sin \theta \)
- \( z = z \)
- \( x^2 + y^2 = r^2 \)
Substitute \( x^2 + y^2 = r^2 \) into the equation:
\[ z = 3 - r^2 \]
Thus, the equation for the paraboloid in cylindrical coordinates is \( z = 3 - r^2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F99d15f92-0bff-4b4d-a47e-2ac33d144271%2Fde9fa5ef-9031-4c4b-a018-ac23cdf40281%2Fvrkmxyk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem:**
Find an equation for the paraboloid \( z = 3 - (x^2 + y^2) \) in cylindrical coordinates.
**Solution:**
To express the given equation in cylindrical coordinates, recall that in cylindrical coordinates, the relationships are:
- \( x = r \cos \theta \)
- \( y = r \sin \theta \)
- \( z = z \)
- \( x^2 + y^2 = r^2 \)
Substitute \( x^2 + y^2 = r^2 \) into the equation:
\[ z = 3 - r^2 \]
Thus, the equation for the paraboloid in cylindrical coordinates is \( z = 3 - r^2 \).
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