Recognize a vector field in a plane Sketch the vector field. x | y || F(x, y) = (1/2, − 4 ) 2 2 02 -2 0 2 0-2 0 2 2 -2 2 -2 -2 -2
Recognize a vector field in a plane Sketch the vector field. x | y || F(x, y) = (1/2, − 4 ) 2 2 02 -2 0 2 0-2 0 2 2 -2 2 -2 -2 -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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Question
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### Recognize a Vector Field in a Plane
**Objective:** Sketch the vector field.
**Given:**
Vector Field:
\[
\mathbf{F}(x, y) = \left\langle \frac{y}{2}, -\frac{x}{2} \right\rangle
\]
| \(x\) | \(y\) | \(\mathbf{F}(x, y)\) |
|:-------:|:-------:|:----------------------:|
| 2 | 0 | |
| 0 | 2 | |
| -2 | 0 | |
| 0 | -2 | |
| 2 | 2 | |
| -2 | 2 | |
| -2 | -2 | |
| 2 | -2 | |
**Instructions:**
- Fill in the vector field values \( \mathbf{F}(x, y) \) in the table.
- Draw the corresponding vectors on the grid.
**Grid Description:**
- A 9x9 grid with axes labeled from -4 to 4 on both axes.
- The button "Clear All" allows you to erase any drawings.
- A drawing tool is available at the bottom labeled "Draw."
**Approach:**
1. Compute \( \mathbf{F}(x, y) \) for each \((x, y)\) pair in the table.
2. Use the vector \(\left\langle \frac{y}{2}, -\frac{x}{2} \right\rangle\) to determine direction and magnitude.
3. Plot each vector on the grid according to its starting point \((x, y)\).
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Transcribed Image Text:---
### Recognize a Vector Field in a Plane
**Objective:** Sketch the vector field.
**Given:**
Vector Field:
\[
\mathbf{F}(x, y) = \left\langle \frac{y}{2}, -\frac{x}{2} \right\rangle
\]
| \(x\) | \(y\) | \(\mathbf{F}(x, y)\) |
|:-------:|:-------:|:----------------------:|
| 2 | 0 | |
| 0 | 2 | |
| -2 | 0 | |
| 0 | -2 | |
| 2 | 2 | |
| -2 | 2 | |
| -2 | -2 | |
| 2 | -2 | |
**Instructions:**
- Fill in the vector field values \( \mathbf{F}(x, y) \) in the table.
- Draw the corresponding vectors on the grid.
**Grid Description:**
- A 9x9 grid with axes labeled from -4 to 4 on both axes.
- The button "Clear All" allows you to erase any drawings.
- A drawing tool is available at the bottom labeled "Draw."
**Approach:**
1. Compute \( \mathbf{F}(x, y) \) for each \((x, y)\) pair in the table.
2. Use the vector \(\left\langle \frac{y}{2}, -\frac{x}{2} \right\rangle\) to determine direction and magnitude.
3. Plot each vector on the grid according to its starting point \((x, y)\).
---
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