# 2: For the vector field F = xi a) Plot the vector field on the xry Cartesian coordinate system b) Find the divergence of the vector field. c) Find the curl of the vector field.
# 2: For the vector field F = xi a) Plot the vector field on the xry Cartesian coordinate system b) Find the divergence of the vector field. c) Find the curl of the vector field.
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
![### Vector Field Analysis
#### Problem Statement
For the vector field \( \mathbf{F} = x \mathbf{i} \):
a) Plot the vector field on the \( xy \) Cartesian coordinate system.
b) Find the divergence of the vector field.
c) Find the curl of the vector field.
#### Solution:
##### a) Plotting the Vector Field
To plot the vector field \( \mathbf{F} = x \mathbf{i} \), we need to represent the vectors at different points in the \( xy \) plane. The vector \( \mathbf{F} = x \mathbf{i} \) means that at a point \((x,y)\), the vector has a magnitude equal to \( x \) and points in the positive \( x \)-direction (since it's scaled by the unit vector \( \mathbf{i} \)).
1. At \((1, 0)\), \( \mathbf{F} = \mathbf{i} \) (a vector of length 1 in the \( x \)-direction).
2. At \((2, 0)\), \( \mathbf{F} = 2 \mathbf{i} \) (a vector of length 2 in the \( x \)-direction).
3. At \((-1, 0)\), \( \mathbf{F} = -\mathbf{i} \) (a vector of length 1 in the negative \( x \)-direction).
4. At any point \((x, y)\), the vector is \( x \mathbf{i} \), independent of \( y \).
This creates a pattern where vectors at points with larger \( x \)-coordinates are longer, and the vectors always point horizontally in the \( x \)-direction.
##### b) Finding the Divergence of the Vector Field
The divergence of a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} \) is given by:
\[ \nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} \]
For \( \mathbf{F} = x \mathbf{i} \):
- \( P = x \)
- \( Q = 0 \)
So,
\[ \nabla \cdot \mathbf{F} = \frac{\partial x}{\partial x} + \frac{\partial 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd11254da-7ef5-43d6-a5d1-60e504f682d9%2F1baa0a23-d07e-4d65-b532-d5d3d076e9c8%2Fuixvvt_processed.png&w=3840&q=75)
Transcribed Image Text:### Vector Field Analysis
#### Problem Statement
For the vector field \( \mathbf{F} = x \mathbf{i} \):
a) Plot the vector field on the \( xy \) Cartesian coordinate system.
b) Find the divergence of the vector field.
c) Find the curl of the vector field.
#### Solution:
##### a) Plotting the Vector Field
To plot the vector field \( \mathbf{F} = x \mathbf{i} \), we need to represent the vectors at different points in the \( xy \) plane. The vector \( \mathbf{F} = x \mathbf{i} \) means that at a point \((x,y)\), the vector has a magnitude equal to \( x \) and points in the positive \( x \)-direction (since it's scaled by the unit vector \( \mathbf{i} \)).
1. At \((1, 0)\), \( \mathbf{F} = \mathbf{i} \) (a vector of length 1 in the \( x \)-direction).
2. At \((2, 0)\), \( \mathbf{F} = 2 \mathbf{i} \) (a vector of length 2 in the \( x \)-direction).
3. At \((-1, 0)\), \( \mathbf{F} = -\mathbf{i} \) (a vector of length 1 in the negative \( x \)-direction).
4. At any point \((x, y)\), the vector is \( x \mathbf{i} \), independent of \( y \).
This creates a pattern where vectors at points with larger \( x \)-coordinates are longer, and the vectors always point horizontally in the \( x \)-direction.
##### b) Finding the Divergence of the Vector Field
The divergence of a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} \) is given by:
\[ \nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} \]
For \( \mathbf{F} = x \mathbf{i} \):
- \( P = x \)
- \( Q = 0 \)
So,
\[ \nabla \cdot \mathbf{F} = \frac{\partial x}{\partial x} + \frac{\partial 0
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