Given that V= pz cos o, find VV and V²V.
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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Question
![**Problem:**
Given that \( V = \rho^2 z \cos \phi \), find \( \nabla V \) and \( \nabla^2 V \).
**Solution:**
To solve this problem, we'll calculate the gradient and the Laplacian of the given function \( V \).
1. **Gradient (\( \nabla V \)):**
The gradient of a scalar field \( V \) in cylindrical coordinates (\( \rho, \phi, z \)) is given by:
\[
\nabla V = \left( \frac{\partial V}{\partial \rho}, \frac{1}{\rho} \frac{\partial V}{\partial \phi}, \frac{\partial V}{\partial z} \right)
\]
- Partial derivative with respect to \( \rho \):
\[
\frac{\partial V}{\partial \rho} = 2\rho z \cos \phi
\]
- Partial derivative with respect to \( \phi \):
\[
\frac{\partial V}{\partial \phi} = -\rho^2 z \sin \phi
\]
Hence,
\[
\frac{1}{\rho} \frac{\partial V}{\partial \phi} = -\rho z \sin \phi
\]
- Partial derivative with respect to \( z \):
\[
\frac{\partial V}{\partial z} = \rho^2 \cos \phi
\]
Thus, the gradient of \( V \) is:
\[
\nabla V = \left( 2\rho z \cos \phi, -\rho z \sin \phi, \rho^2 \cos \phi \right)
\]
2. **Laplacian (\( \nabla^2 V \)):**
The Laplacian of a scalar field \( V \) in cylindrical coordinates is given by:
\[
\nabla^2 V = \frac{1}{\rho} \frac{\partial}{\partial \rho}\left(\rho \frac{\partial V}{\partial \rho}\right) + \frac{1}{\rho^2} \frac{\partial^2 V}{\partial \phi^2} + \frac{\partial^2 V}{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4bf62d6d-db44-481f-809c-2b5fe2fdf00c%2F48cedc31-0d60-4620-96d8-99a6bdc8b704%2Fwieirle_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem:**
Given that \( V = \rho^2 z \cos \phi \), find \( \nabla V \) and \( \nabla^2 V \).
**Solution:**
To solve this problem, we'll calculate the gradient and the Laplacian of the given function \( V \).
1. **Gradient (\( \nabla V \)):**
The gradient of a scalar field \( V \) in cylindrical coordinates (\( \rho, \phi, z \)) is given by:
\[
\nabla V = \left( \frac{\partial V}{\partial \rho}, \frac{1}{\rho} \frac{\partial V}{\partial \phi}, \frac{\partial V}{\partial z} \right)
\]
- Partial derivative with respect to \( \rho \):
\[
\frac{\partial V}{\partial \rho} = 2\rho z \cos \phi
\]
- Partial derivative with respect to \( \phi \):
\[
\frac{\partial V}{\partial \phi} = -\rho^2 z \sin \phi
\]
Hence,
\[
\frac{1}{\rho} \frac{\partial V}{\partial \phi} = -\rho z \sin \phi
\]
- Partial derivative with respect to \( z \):
\[
\frac{\partial V}{\partial z} = \rho^2 \cos \phi
\]
Thus, the gradient of \( V \) is:
\[
\nabla V = \left( 2\rho z \cos \phi, -\rho z \sin \phi, \rho^2 \cos \phi \right)
\]
2. **Laplacian (\( \nabla^2 V \)):**
The Laplacian of a scalar field \( V \) in cylindrical coordinates is given by:
\[
\nabla^2 V = \frac{1}{\rho} \frac{\partial}{\partial \rho}\left(\rho \frac{\partial V}{\partial \rho}\right) + \frac{1}{\rho^2} \frac{\partial^2 V}{\partial \phi^2} + \frac{\partial^2 V}{
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