Let r = xi + yj + zk and r = |rl. If F = find div(F). (Enter your answer in terms of r and p.) p' div(F) = Is there a value of p for which div(F) = 0? (If an answer does not exist, enter DNE.) P =
Let r = xi + yj + zk and r = |rl. If F = find div(F). (Enter your answer in terms of r and p.) p' div(F) = Is there a value of p for which div(F) = 0? (If an answer does not exist, enter DNE.) P =
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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![### Educational Website Content
#### Vector Field and Divergence Calculation
Let's consider the vector \(\mathbf{r}\) defined as:
\[ \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \]
where \( \mathbf{r} \) is a position vector in 3D space. The magnitude of \(\mathbf{r}\), denoted as \( r \), is:
\[ r = |\mathbf{r}| \]
Now, given the vector field:
\[ \mathbf{F} = \frac{\mathbf{r}}{r^p} \]
we aim to find the divergence of \(\mathbf{F}\), denoted as \(\text{div}(\mathbf{F})\).
##### Step-by-Step Problem Breakdown:
1. **Calculate the Divergence of \(\mathbf{F}\):**
Determine \(\text{div}(\mathbf{F})\) by entering your answer in terms of \(r\) and \(p\):
\[
\text{div}(\mathbf{F}) = \boxed{}
\]
2. **Find the Value of \(p\) for Zero Divergence:**
Determine if there exists a value of \( p \) for which \(\text{div}(\mathbf{F}) = 0\). If no such value exists, enter "DNE" (Does Not Exist):
\[
p = \boxed{}
\]
#### Explanation and Computation
1. Begin by expressing the position vector \(\mathbf{r}\) and its magnitude \(r\).
2. Substitute the expressions into the given vector field \(\mathbf{F}\).
3. Use the definition of divergence for a vector field \(\mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}\):
\[
\text{div}(\mathbf{F}) = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}
\]
4. Simplify the expression to find \(\text{div}(\mathbf{F})\) in terms of \(r\) and \(p\).
5. Solve for \(p\) if \(\text{div}(\mathbf{F}) = 0\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8db4a5d1-5f92-40a1-9cd8-b619f03354b1%2Fa6985ae8-5828-409c-b433-771238eb609d%2Fmjtjb6e_processed.png&w=3840&q=75)
Transcribed Image Text:### Educational Website Content
#### Vector Field and Divergence Calculation
Let's consider the vector \(\mathbf{r}\) defined as:
\[ \mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \]
where \( \mathbf{r} \) is a position vector in 3D space. The magnitude of \(\mathbf{r}\), denoted as \( r \), is:
\[ r = |\mathbf{r}| \]
Now, given the vector field:
\[ \mathbf{F} = \frac{\mathbf{r}}{r^p} \]
we aim to find the divergence of \(\mathbf{F}\), denoted as \(\text{div}(\mathbf{F})\).
##### Step-by-Step Problem Breakdown:
1. **Calculate the Divergence of \(\mathbf{F}\):**
Determine \(\text{div}(\mathbf{F})\) by entering your answer in terms of \(r\) and \(p\):
\[
\text{div}(\mathbf{F}) = \boxed{}
\]
2. **Find the Value of \(p\) for Zero Divergence:**
Determine if there exists a value of \( p \) for which \(\text{div}(\mathbf{F}) = 0\). If no such value exists, enter "DNE" (Does Not Exist):
\[
p = \boxed{}
\]
#### Explanation and Computation
1. Begin by expressing the position vector \(\mathbf{r}\) and its magnitude \(r\).
2. Substitute the expressions into the given vector field \(\mathbf{F}\).
3. Use the definition of divergence for a vector field \(\mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}\):
\[
\text{div}(\mathbf{F}) = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}
\]
4. Simplify the expression to find \(\text{div}(\mathbf{F})\) in terms of \(r\) and \(p\).
5. Solve for \(p\) if \(\text{div}(\mathbf{F}) = 0\
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