Use the divergence theorem to calculate the flux of F = (x - 8y)i + (y – 4z)j + (z - 8x)k out of the unit sphere.
Use the divergence theorem to calculate the flux of F = (x - 8y)i + (y – 4z)j + (z - 8x)k out of the unit sphere.
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
**Objective:** Use the divergence theorem to calculate the flux of the vector field **F** through the surface of the unit sphere.
**Vector Field:**
\[ \mathbf{F} = (x - 8y)\mathbf{i} + (y - 4z)\mathbf{j} + (z - 8x)\mathbf{k} \]
**Surface:** The unit sphere.
### Explanation of the Divergence Theorem
The divergence theorem, also known as Gauss's theorem or Ostrogradsky’s theorem, relates the flow (or flux) of a vector field through a closed surface to the divergence of the vector field inside the volume enclosed by the surface.
**Mathematically:**
\[ \iint_{\partial V} \mathbf{F} \cdot d\mathbf{S} = \iiint_{V} (\nabla \cdot \mathbf{F}) \, dV \]
Where:
- \(\iint_{\partial V} \mathbf{F} \cdot d\mathbf{S}\) is the flux of **F** across the boundary surface \(\partial V\).
- \(\iiint_{V} (\nabla \cdot \mathbf{F}) \, dV\) is the triple integral of the divergence of **F** over the volume \(V\).
### Steps to Solve
1. **Calculate the Divergence** \(\nabla \cdot \mathbf{F}\):
\[\nabla \cdot \mathbf{F} = \frac{\partial}{\partial x}(x - 8y) + \frac{\partial}{\partial y}(y - 4z) + \frac{\partial}{\partial z}(z - 8x)\]
2. **Set Up the Volume Integral** over the unit sphere:
\[\iiint_{V} (\nabla \cdot \mathbf{F}) \, dV\]
3. **Evaluate** the volume integral, which is equal to the flux through the unit sphere by the divergence theorem.
This explanation and calculation help demonstrate one of the powerful applications of vector calculus in solving physics and engineering problems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd5b6e13e-b165-4b79-92da-bc958eab9f6d%2F505463bd-7ea9-41f3-867f-4be074e0ec92%2Fd2dh84k_processed.png&w=3840&q=75)
Transcribed Image Text:## Problem Statement
**Objective:** Use the divergence theorem to calculate the flux of the vector field **F** through the surface of the unit sphere.
**Vector Field:**
\[ \mathbf{F} = (x - 8y)\mathbf{i} + (y - 4z)\mathbf{j} + (z - 8x)\mathbf{k} \]
**Surface:** The unit sphere.
### Explanation of the Divergence Theorem
The divergence theorem, also known as Gauss's theorem or Ostrogradsky’s theorem, relates the flow (or flux) of a vector field through a closed surface to the divergence of the vector field inside the volume enclosed by the surface.
**Mathematically:**
\[ \iint_{\partial V} \mathbf{F} \cdot d\mathbf{S} = \iiint_{V} (\nabla \cdot \mathbf{F}) \, dV \]
Where:
- \(\iint_{\partial V} \mathbf{F} \cdot d\mathbf{S}\) is the flux of **F** across the boundary surface \(\partial V\).
- \(\iiint_{V} (\nabla \cdot \mathbf{F}) \, dV\) is the triple integral of the divergence of **F** over the volume \(V\).
### Steps to Solve
1. **Calculate the Divergence** \(\nabla \cdot \mathbf{F}\):
\[\nabla \cdot \mathbf{F} = \frac{\partial}{\partial x}(x - 8y) + \frac{\partial}{\partial y}(y - 4z) + \frac{\partial}{\partial z}(z - 8x)\]
2. **Set Up the Volume Integral** over the unit sphere:
\[\iiint_{V} (\nabla \cdot \mathbf{F}) \, dV\]
3. **Evaluate** the volume integral, which is equal to the flux through the unit sphere by the divergence theorem.
This explanation and calculation help demonstrate one of the powerful applications of vector calculus in solving physics and engineering problems.
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