Derive the 2D Laplacian operator in polar coordinates, using the following relationship between x, y, and R, 0: x = = R cos(0) y = Rsin(0)
Derive the 2D Laplacian operator in polar coordinates, using the following relationship between x, y, and R, 0: x = = R cos(0) y = Rsin(0)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Title: Derivation of the 2D Laplacian Operator in Polar Coordinates**
**Introduction:**
In this section, we will derive the 2D Laplacian operator in polar coordinates. The transformation from Cartesian to polar coordinates involves the following relationships between \(x\), \(y\), radius \(R\), and angle \(\theta\):
\[ x = R \cos(\theta) \]
\[ y = R \sin(\theta) \]
**Objective:**
Our goal is to express the Laplacian operator, commonly used in various fields such as physics and engineering, in terms of \(R\) and \(\theta\). This derivation is essential for solving problems that have circular or radial symmetry.
**Steps to Derivation:**
1. **Express Cartesian Coordinates in Terms of Polar Coordinates:**
\[
x = R \cos(\theta)
\]
\[
y = R \sin(\theta)
\]
2. **Substitute These Relationships into the Standard 2D Laplacian Formula in Cartesian Coordinates.**
3. **Apply Chain Rule to Account for the Change of Variables.**
4. **Simplify the Result to Obtain the Laplacian in Polar Form.**
This transformation is crucial for simplifying the analysis of problems involving circular geometries or radial fields. The detailed steps and mathematical manipulations are typically presented in advanced calculus or mathematical physics courses.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff6e2f997-9120-4975-9388-a1bc7e4c3a16%2Facf71a09-6b6b-4d23-8fc8-6e10291053a3%2F8u39kcb_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Derivation of the 2D Laplacian Operator in Polar Coordinates**
**Introduction:**
In this section, we will derive the 2D Laplacian operator in polar coordinates. The transformation from Cartesian to polar coordinates involves the following relationships between \(x\), \(y\), radius \(R\), and angle \(\theta\):
\[ x = R \cos(\theta) \]
\[ y = R \sin(\theta) \]
**Objective:**
Our goal is to express the Laplacian operator, commonly used in various fields such as physics and engineering, in terms of \(R\) and \(\theta\). This derivation is essential for solving problems that have circular or radial symmetry.
**Steps to Derivation:**
1. **Express Cartesian Coordinates in Terms of Polar Coordinates:**
\[
x = R \cos(\theta)
\]
\[
y = R \sin(\theta)
\]
2. **Substitute These Relationships into the Standard 2D Laplacian Formula in Cartesian Coordinates.**
3. **Apply Chain Rule to Account for the Change of Variables.**
4. **Simplify the Result to Obtain the Laplacian in Polar Form.**
This transformation is crucial for simplifying the analysis of problems involving circular geometries or radial fields. The detailed steps and mathematical manipulations are typically presented in advanced calculus or mathematical physics courses.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
