Find the divergence of the vector field F at the given point. F(x, y, z) xyzi + xz²j + 4yz²k; (5, 2, 1)
Find the divergence of the vector field F at the given point. F(x, y, z) xyzi + xz²j + 4yz²k; (5, 2, 1)
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 Statement:**
Find the divergence of the vector field **F** at the given point.
\[ \mathbf{F}(x, y, z) = xyz \, \mathbf{i} + xz^2 \, \mathbf{j} + 4yz^2 \, \mathbf{k}; \quad (5, 2, 1) \]
**Explanation:**
- The vector field **F** is given in terms of its components with the basis vectors **i**, **j**, **k**.
- The components of the vector field are:
- \( xyz \) along the **i** direction
- \( xz^2 \) along the **j** direction
- \( 4yz^2 \) along the **k** direction
- The point at which the divergence is to be calculated is \((5, 2, 1)\).
**Divergence Concept:**
The divergence of a vector field \(\mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k}\) is calculated as:
\[
\nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}
\]
**Steps to Solve:**
1. Take the partial derivative of \( xyz \) with respect to \( x \).
2. Take the partial derivative of \( xz^2 \) with respect to \( y \).
3. Take the partial derivative of \( 4yz^2 \) with respect to \( z \).
4. Substitute the point \((5, 2, 1)\) into the resulting expression to find the numerical value of the divergence.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb9ffc63-b6b6-4fe8-8e4f-4fc825094d26%2F553574a0-fb02-4481-b542-b152b3247f95%2F6u6js4_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the divergence of the vector field **F** at the given point.
\[ \mathbf{F}(x, y, z) = xyz \, \mathbf{i} + xz^2 \, \mathbf{j} + 4yz^2 \, \mathbf{k}; \quad (5, 2, 1) \]
**Explanation:**
- The vector field **F** is given in terms of its components with the basis vectors **i**, **j**, **k**.
- The components of the vector field are:
- \( xyz \) along the **i** direction
- \( xz^2 \) along the **j** direction
- \( 4yz^2 \) along the **k** direction
- The point at which the divergence is to be calculated is \((5, 2, 1)\).
**Divergence Concept:**
The divergence of a vector field \(\mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k}\) is calculated as:
\[
\nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}
\]
**Steps to Solve:**
1. Take the partial derivative of \( xyz \) with respect to \( x \).
2. Take the partial derivative of \( xz^2 \) with respect to \( y \).
3. Take the partial derivative of \( 4yz^2 \) with respect to \( z \).
4. Substitute the point \((5, 2, 1)\) into the resulting expression to find the numerical value of the divergence.
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