- Here is a graph of the vector field F(r, y) = r²i+ y°j. 2 (a) Calculate div F. (b) For which points (r, y) do we have div F = 0? How can we visualize this? %3D
- Here is a graph of the vector field F(r, y) = r²i+ y°j. 2 (a) Calculate div F. (b) For which points (r, y) do we have div F = 0? How can we visualize this? %3D
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
Topic Video
Question
100%
REFER TO IMAGE
This graph displays arrows indicating the direction and magnitude of the vector field at different points in the plane. The arrows point outward from the origin, with their lengths increasing as they move away from the origin, particularly emphasizing the behavior of \( \mathbf{F}(x, y) \).
#### Problems and Solutions
(a) **Calculate \(\operatorname{div} \mathbf{F}\):**
To find the divergence of the vector field \( \mathbf{F}(x, y) = x^2 \hat{\mathbf{i}} + y^2 \hat{\mathbf{j}} \):
\[
\operatorname{div} \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y}
\]
Here, \( F_1 = x^2 \) and \( F_2 = y^2 \).
\[
\operatorname{div} \mathbf{F} = \frac{\partial x^2}{\partial x} + \frac{\partial y^2}{\partial y} = 2x + 2y
\]
(b) **For which points \((x, y)\) do we have \(\operatorname{div} \mathbf{F} = 0\)? How can we visualize this?**
Setting the divergence to zero, we get:
\[
2x + 2y = 0 \implies x + y = 0
\]
These points lie along the line \( y = -x \).
(c) **For which points \((x, y)\) do we have \(\operatorname{div} \mathbf{F} > 0\)? How can we visualize this?**
For the divergence to be positive:
\[
2x + 2y > 0 \implies x +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F191441f0-66db-457d-b504-dd20fe1485fa%2Fed62d99f-407e-4633-95c4-0470e00fb172%2Fldp4ne5_processed.png&w=3840&q=75)
Transcribed Image Text:---
### Understanding the Divergence of a Vector Field
In this section, we will explore the concept of divergence in the context of a given vector field. Let’s analyze the vector field \( \mathbf{F}(x, y) = x^2 \hat{\mathbf{i}} + y^2 \hat{\mathbf{j}} \).
#### Vector Field Visualization
Below is a graph representing this vector field:
This graph displays arrows indicating the direction and magnitude of the vector field at different points in the plane. The arrows point outward from the origin, with their lengths increasing as they move away from the origin, particularly emphasizing the behavior of \( \mathbf{F}(x, y) \).
#### Problems and Solutions
(a) **Calculate \(\operatorname{div} \mathbf{F}\):**
To find the divergence of the vector field \( \mathbf{F}(x, y) = x^2 \hat{\mathbf{i}} + y^2 \hat{\mathbf{j}} \):
\[
\operatorname{div} \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y}
\]
Here, \( F_1 = x^2 \) and \( F_2 = y^2 \).
\[
\operatorname{div} \mathbf{F} = \frac{\partial x^2}{\partial x} + \frac{\partial y^2}{\partial y} = 2x + 2y
\]
(b) **For which points \((x, y)\) do we have \(\operatorname{div} \mathbf{F} = 0\)? How can we visualize this?**
Setting the divergence to zero, we get:
\[
2x + 2y = 0 \implies x + y = 0
\]
These points lie along the line \( y = -x \).
(c) **For which points \((x, y)\) do we have \(\operatorname{div} \mathbf{F} > 0\)? How can we visualize this?**
For the divergence to be positive:
\[
2x + 2y > 0 \implies x +
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 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.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,
