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

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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

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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

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