4. Consider the vector field F(r, y, z) = (x, y, z). (a) Sketch a picture of F. (b) Calculate div F. How can we visualize div F in terms of the vector field? (c) Suppose that F represents a fluid flow in 3-d space. Imagine a small sphere or cube placed at the point (3, 0, 4). Is more fluid flowing into the object or out of it?
4. Consider the vector field F(r, y, z) = (x, y, z). (a) Sketch a picture of F. (b) Calculate div F. How can we visualize div F in terms of the vector field? (c) Suppose that F represents a fluid flow in 3-d space. Imagine a small sphere or cube placed at the point (3, 0, 4). Is more fluid flowing into the object or out of it?
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
REFER TO IMAGE
![**Problem 4: Vector Field Analysis**
Consider the vector field \(\mathbf{F}(x, y, z) = \langle x, y, z \rangle\).
**(a) Sketch a picture of \(\mathbf{F}\).**
(Sketch Description: In this section, you need to create a vector field diagram for \(\mathbf{F}(x, y, z) = \langle x, y, z \rangle\). This means that at any point \((x, y, z)\) in space, the vector \(\mathbf{F}\) points directly away from the origin and has a magnitude equal to the distance from the origin. The length of the vectors increases as you move further from the origin.)
**(b) Calculate \(\text{div} \mathbf{F}\). How can we visualize \(\text{div} \mathbf{F}\) in terms of the vector field?**
To calculate the divergence of \(\mathbf{F}\), use the formula for the divergence of a vector field:
\[
\text{div} \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}
\]
Given \(\mathbf{F}(x, y, z) = \langle x, y, z \rangle\), the components are \(F_1 = x\), \(F_2 = y\), and \(F_3 = z\). Taking the partial derivatives:
\[
\frac{\partial F_1}{\partial x} = 1, \quad \frac{\partial F_2}{\partial y} = 1, \quad \frac{\partial F_3}{\partial z} = 1
\]
Thus,
\[
\text{div} \mathbf{F} = 1 + 1 + 1 = 3
\]
The divergence of \(\mathbf{F}\) is a constant value of 3. This means that, at any point in space, the vector field \(\mathbf{F}\) represents a uniform source, indicating that fluid is flowing outward from each point in space with a constant rate.
**(c) Suppose that \(\mathbf{F}\) represents a fluid flow in 3-D](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F191441f0-66db-457d-b504-dd20fe1485fa%2F54926a54-2576-46f5-b036-4d53f775d44e%2Fxqiwqhb_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 4: Vector Field Analysis**
Consider the vector field \(\mathbf{F}(x, y, z) = \langle x, y, z \rangle\).
**(a) Sketch a picture of \(\mathbf{F}\).**
(Sketch Description: In this section, you need to create a vector field diagram for \(\mathbf{F}(x, y, z) = \langle x, y, z \rangle\). This means that at any point \((x, y, z)\) in space, the vector \(\mathbf{F}\) points directly away from the origin and has a magnitude equal to the distance from the origin. The length of the vectors increases as you move further from the origin.)
**(b) Calculate \(\text{div} \mathbf{F}\). How can we visualize \(\text{div} \mathbf{F}\) in terms of the vector field?**
To calculate the divergence of \(\mathbf{F}\), use the formula for the divergence of a vector field:
\[
\text{div} \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}
\]
Given \(\mathbf{F}(x, y, z) = \langle x, y, z \rangle\), the components are \(F_1 = x\), \(F_2 = y\), and \(F_3 = z\). Taking the partial derivatives:
\[
\frac{\partial F_1}{\partial x} = 1, \quad \frac{\partial F_2}{\partial y} = 1, \quad \frac{\partial F_3}{\partial z} = 1
\]
Thus,
\[
\text{div} \mathbf{F} = 1 + 1 + 1 = 3
\]
The divergence of \(\mathbf{F}\) is a constant value of 3. This means that, at any point in space, the vector field \(\mathbf{F}\) represents a uniform source, indicating that fluid is flowing outward from each point in space with a constant rate.
**(c) Suppose that \(\mathbf{F}\) represents a fluid flow in 3-D
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
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 3 steps with 3 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
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](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
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…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
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…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Mathematics For Machine Technology](https://www.bartleby.com/isbn_cover_images/9781337798310/9781337798310_smallCoverImage.jpg)
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
![Basic Technical Mathematics](https://www.bartleby.com/isbn_cover_images/9780134437705/9780134437705_smallCoverImage.gif)
![Topology](https://www.bartleby.com/isbn_cover_images/9780134689517/9780134689517_smallCoverImage.gif)