Use the Divergence Theorem to evaluate / F. ds where and S is the boundary of the sphere x² + y² + z² = 4 oriented by the outward normal. The surface integral equals F = (10x4, 3yz, -40x³z)
Use the Divergence Theorem to evaluate / F. ds where and S is the boundary of the sphere x² + y² + z² = 4 oriented by the outward normal. The surface integral equals F = (10x4, 3yz, -40x³z)
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Application of the Divergence Theorem
In this example, we will use the Divergence Theorem to evaluate the surface integral of a vector field \(\mathbf{F}\) over a closed surface \(S\).
#### Problem Statement
Evaluate the surface integral:
\[
\iint_S \mathbf{F} \cdot d\mathbf{S}
\]
where:
\[
\mathbf{F} = \langle 10x^4, 3yz^6, -40x^3z \rangle
\]
and \(S\) is the boundary of the sphere defined by:
\[
x^2 + y^2 + z^2 = 4
\]
The surface \(S\) is oriented by the outward normal.
#### Using the Divergence Theorem
The Divergence Theorem states:
\[
\iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_V (\nabla \cdot \mathbf{F}) \, dV
\]
where \(V\) is the volume enclosed by the surface \(S\).
1. **Compute the Divergence of \(\mathbf{F}\)**:
The divergence of \(\mathbf{F}\), denoted as \(\nabla \cdot \mathbf{F}\), is:
\[
\nabla \cdot \mathbf{F} = \frac{\partial}{\partial x}(10x^4) + \frac{\partial}{\partial y}(3yz^6) + \frac{\partial}{\partial z}(-40x^3z)
\]
2. **Calculate each partial derivative**:
\[
\frac{\partial}{\partial x}(10x^4) = 40x^3
\]
\[
\frac{\partial}{\partial y}(3yz^6) = 3z^6
\]
\[
\frac{\partial}{\partial z}(-40x^3z) = -40x^3
\]
3. **Sum the partial derivatives**:
\[
\nabla \cdot \mathbf{F} = 40x^3 + 3z^6 - 40x^3 = 3z^6
\]
4. **Set up the volume integral**:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf673b33-84d3-4207-a3d8-77b439e8ab65%2Fe6e8aa89-ddf1-4880-a332-440a794a5131%2Flkuy6ro_processed.png&w=3840&q=75)
Transcribed Image Text:### Application of the Divergence Theorem
In this example, we will use the Divergence Theorem to evaluate the surface integral of a vector field \(\mathbf{F}\) over a closed surface \(S\).
#### Problem Statement
Evaluate the surface integral:
\[
\iint_S \mathbf{F} \cdot d\mathbf{S}
\]
where:
\[
\mathbf{F} = \langle 10x^4, 3yz^6, -40x^3z \rangle
\]
and \(S\) is the boundary of the sphere defined by:
\[
x^2 + y^2 + z^2 = 4
\]
The surface \(S\) is oriented by the outward normal.
#### Using the Divergence Theorem
The Divergence Theorem states:
\[
\iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_V (\nabla \cdot \mathbf{F}) \, dV
\]
where \(V\) is the volume enclosed by the surface \(S\).
1. **Compute the Divergence of \(\mathbf{F}\)**:
The divergence of \(\mathbf{F}\), denoted as \(\nabla \cdot \mathbf{F}\), is:
\[
\nabla \cdot \mathbf{F} = \frac{\partial}{\partial x}(10x^4) + \frac{\partial}{\partial y}(3yz^6) + \frac{\partial}{\partial z}(-40x^3z)
\]
2. **Calculate each partial derivative**:
\[
\frac{\partial}{\partial x}(10x^4) = 40x^3
\]
\[
\frac{\partial}{\partial y}(3yz^6) = 3z^6
\]
\[
\frac{\partial}{\partial z}(-40x^3z) = -40x^3
\]
3. **Sum the partial derivatives**:
\[
\nabla \cdot \mathbf{F} = 40x^3 + 3z^6 - 40x^3 = 3z^6
\]
4. **Set up the volume integral**:
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