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Sketch the vector field. X 2 0 ० -2 ० y -2 2 ० 0 -4 -3 2 2 -2 N -2 -2 2 -2 -2 X Y F(x, y) = ( − 1 2 ¹ 2) (T, Y) 2' <-1,0> <0,1> <1,0> <0,-1> <-1,1> <1,1> <1,-1> <-1,-1> 3 2 1 -1 -2 -3 1 2 > > > ४४ > > J ४ ४४ 3 O

Sketch the vector field. X 2 0 ० -2 ० y -2 2 ० 0 -4 -3 2 2 -2 N -2 -2 2 -2 -2 X Y F(x, y) = ( − 1 2 ¹ 2) (T, Y) 2' <-1,0> <0,1> <1,0> <0,-1> <-1,1> <1,1> <1,-1> <-1,-1> 3 2 1 -1 -2 -3 1 2 > > > ४४ > > J ४ ४४ 3 O

Advanced Engineering Mathematics
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

6.1.1

**Vector Field Exploration** **Objective:** Sketch the vector field defined by \(\vec{F}(x, y) = \left\langle -\frac{x}{2}, \frac{y}{2} \right\rangle\). **Data Table:** | \(x\) | \(y\) | \(\vec{F}(x, y) = \left\langle -\frac{x}{2}, \frac{y}{2} \right\rangle\) | Vector | Confirmation | |--------|--------|------------------------------------------------------------------|-------------|--------------| | 2 | 0 | \(\left\langle -1, 0 \right\rangle\) | <1,0> | ✔️ | | 0 | 2 | \(\left\langle 0, 1 \right\rangle\) | <0,1> | ✔️ | | -2 | 0 | \(\left\langle 1, 0 \right\rangle\) | <1,0> | ✔️ | | 0 | -2 | \(\left\langle 0, -1 \right\rangle\) | <0,-1> | ✔️ | | 2 | 2 | \(\left\langle -1, 1 \right\rangle\) | <1,1> | ✔️ | | -2 | 2 | \(\left\langle 1, 1 \right\rangle\) | <1,1> | ✔️ | | -2 | -2 | \(\left\langle 1, -1 \right\rangle\) | <1,-1> | ✔️ | | 2 | -2 | \(\left\langle -1, -1 \right\rangle\) | <-1,-1> | ✔️ | **Graph Explanation:** The graph is a Cartesian coordinate system ranging from -4 to 4 on both the x-axis and the y-axis, with grid lines at each integer. You can plot the vectors calculated from the table onto this graph to visualize the behavior of the vector field. Each vector originates at its corresponding (x
Transcribed Image Text:**Vector Field Exploration** **Objective:** Sketch the vector field defined by \(\vec{F}(x, y) = \left\langle -\frac{x}{2}, \frac{y}{2} \right\rangle\). **Data Table:** | \(x\) | \(y\) | \(\vec{F}(x, y) = \left\langle -\frac{x}{2}, \frac{y}{2} \right\rangle\) | Vector | Confirmation | |--------|--------|------------------------------------------------------------------|-------------|--------------| | 2 | 0 | \(\left\langle -1, 0 \right\rangle\) | <1,0> | ✔️ | | 0 | 2 | \(\left\langle 0, 1 \right\rangle\) | <0,1> | ✔️ | | -2 | 0 | \(\left\langle 1, 0 \right\rangle\) | <1,0> | ✔️ | | 0 | -2 | \(\left\langle 0, -1 \right\rangle\) | <0,-1> | ✔️ | | 2 | 2 | \(\left\langle -1, 1 \right\rangle\) | <1,1> | ✔️ | | -2 | 2 | \(\left\langle 1, 1 \right\rangle\) | <1,1> | ✔️ | | -2 | -2 | \(\left\langle 1, -1 \right\rangle\) | <1,-1> | ✔️ | | 2 | -2 | \(\left\langle -1, -1 \right\rangle\) | <-1,-1> | ✔️ | **Graph Explanation:** The graph is a Cartesian coordinate system ranging from -4 to 4 on both the x-axis and the y-axis, with grid lines at each integer. You can plot the vectors calculated from the table onto this graph to visualize the behavior of the vector field. Each vector originates at its corresponding (x
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