Find the divergence of the vector field V(x, y, z) = -7exyi – eбxyj + 4e⁹yzk. (Give an exact answer. Use symbolic notation and fractions where needed.) divV =
Find the divergence of the vector field V(x, y, z) = -7exyi – eбxyj + 4e⁹yzk. (Give an exact answer. Use symbolic notation and fractions where needed.) divV =
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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![**Problem: Finding the Divergence of a Vector Field**
Find the divergence of the vector field \( \mathbf{V}(x, y, z) = -7e^{xy}\mathbf{i} - e^{6xy}\mathbf{j} + 4e^{9yz}\mathbf{k} \).
*Instructions:*
- Give an exact answer.
- Use symbolic notation and fractions where needed.
**Solution:**
Calculate the divergence \( \text{div} \mathbf{V} \) of the vector field. The divergence of a vector field \( \mathbf{V} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \) is given by:
\[
\text{div} \mathbf{V} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}
\]
For the given vector field \( \mathbf{V}(x, y, z) = -7e^{xy}\mathbf{i} - e^{6xy}\mathbf{j} + 4e^{9yz}\mathbf{k} \), we identify:
- \( P = -7e^{xy} \)
- \( Q = -e^{6xy} \)
- \( R = 4e^{9yz} \)
Compute each partial derivative:
1. \(\frac{\partial P}{\partial x} = \frac{\partial}{\partial x}(-7e^{xy}) = -7ye^{xy}\)
2. \(\frac{\partial Q}{\partial y} = \frac{\partial}{\partial y}(-e^{6xy}) = -6xe^{6xy}\)
3. \(\frac{\partial R}{\partial z} = \frac{\partial}{\partial z}(4e^{9yz}) = 36ye^{9yz}\)
Thus, the divergence is:
\[
\text{div} \mathbf{V} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} = -7ye^{xy} - 6xe^{6xy} + 36ye^{9yz}
\]
So, your final answer is:
\[
\text{div} \mathbf{V} = -7ye^{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F53419bf8-add0-48cb-b87d-efce75dfc052%2Fdc8f48ed-f058-4755-8171-3d63547305fe%2Fp4gxyk_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem: Finding the Divergence of a Vector Field**
Find the divergence of the vector field \( \mathbf{V}(x, y, z) = -7e^{xy}\mathbf{i} - e^{6xy}\mathbf{j} + 4e^{9yz}\mathbf{k} \).
*Instructions:*
- Give an exact answer.
- Use symbolic notation and fractions where needed.
**Solution:**
Calculate the divergence \( \text{div} \mathbf{V} \) of the vector field. The divergence of a vector field \( \mathbf{V} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \) is given by:
\[
\text{div} \mathbf{V} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}
\]
For the given vector field \( \mathbf{V}(x, y, z) = -7e^{xy}\mathbf{i} - e^{6xy}\mathbf{j} + 4e^{9yz}\mathbf{k} \), we identify:
- \( P = -7e^{xy} \)
- \( Q = -e^{6xy} \)
- \( R = 4e^{9yz} \)
Compute each partial derivative:
1. \(\frac{\partial P}{\partial x} = \frac{\partial}{\partial x}(-7e^{xy}) = -7ye^{xy}\)
2. \(\frac{\partial Q}{\partial y} = \frac{\partial}{\partial y}(-e^{6xy}) = -6xe^{6xy}\)
3. \(\frac{\partial R}{\partial z} = \frac{\partial}{\partial z}(4e^{9yz}) = 36ye^{9yz}\)
Thus, the divergence is:
\[
\text{div} \mathbf{V} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} = -7ye^{xy} - 6xe^{6xy} + 36ye^{9yz}
\]
So, your final answer is:
\[
\text{div} \mathbf{V} = -7ye^{
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