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, 3yz6, -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, 3yz6, -40x³z)
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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![### Using the Divergence Theorem in Multivariable Calculus
In this example, we will use the Divergence Theorem to evaluate the surface integral of a vector field **F** over a closed surface **S**. Specifically, we will compute the following integral:
\[ \iint_S \mathbf{F} \cdot d\mathbf{S} \]
where the vector field \(\mathbf{F}\) is defined as:
\[ \mathbf{F} = \left\langle 10x^4, 3yz^6, -40x^3z \right\rangle \]
and **S** is the boundary of the sphere described by the equation:
\[ x^2 + y^2 + z^2 = 4 \]
### Given Information
- \(\mathbf{F} = \left\langle 10x^4, 3yz^6, -40x^3z \right\rangle\)
- \(S\) is the sphere \(x^2 + y^2 + z^2 = 4\) oriented by the outward normal.
### Objective
To find the value of the surface integral:
\[ \iint_S \mathbf{F} \cdot d\mathbf{S} \]
using the Divergence Theorem.
The Divergence Theorem states:
\[ \iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_V \mathbf{\nabla} \cdot \mathbf{F} \, dV \]
where \(V\) is the volume enclosed by \(S\).
### Applying the Divergence Theorem
To proceed, we need to calculate the divergence of the vector field \(\mathbf{F}\):
\[ \mathbf{\nabla} \cdot \mathbf{F} = \frac{\partial}{\partial x} (10x^4) + \frac{\partial}{\partial y} (3yz^6) + \frac{\partial}{\partial z} (-40x^3z) \]
1. Calculate each partial derivative:
- \(\frac{\partial}{\partial x} (10x^4) = 40x^3\)
- \(\frac{\partial}{\partial y} (3yz^6) = 3z^6\)
- \(\frac{\partial}{\partial z](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf673b33-84d3-4207-a3d8-77b439e8ab65%2F417f171c-eef5-41e7-a225-3d84fbeaae0d%2F7qrduk_processed.png&w=3840&q=75)
Transcribed Image Text:### Using the Divergence Theorem in Multivariable Calculus
In this example, we will use the Divergence Theorem to evaluate the surface integral of a vector field **F** over a closed surface **S**. Specifically, we will compute the following integral:
\[ \iint_S \mathbf{F} \cdot d\mathbf{S} \]
where the vector field \(\mathbf{F}\) is defined as:
\[ \mathbf{F} = \left\langle 10x^4, 3yz^6, -40x^3z \right\rangle \]
and **S** is the boundary of the sphere described by the equation:
\[ x^2 + y^2 + z^2 = 4 \]
### Given Information
- \(\mathbf{F} = \left\langle 10x^4, 3yz^6, -40x^3z \right\rangle\)
- \(S\) is the sphere \(x^2 + y^2 + z^2 = 4\) oriented by the outward normal.
### Objective
To find the value of the surface integral:
\[ \iint_S \mathbf{F} \cdot d\mathbf{S} \]
using the Divergence Theorem.
The Divergence Theorem states:
\[ \iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_V \mathbf{\nabla} \cdot \mathbf{F} \, dV \]
where \(V\) is the volume enclosed by \(S\).
### Applying the Divergence Theorem
To proceed, we need to calculate the divergence of the vector field \(\mathbf{F}\):
\[ \mathbf{\nabla} \cdot \mathbf{F} = \frac{\partial}{\partial x} (10x^4) + \frac{\partial}{\partial y} (3yz^6) + \frac{\partial}{\partial z} (-40x^3z) \]
1. Calculate each partial derivative:
- \(\frac{\partial}{\partial x} (10x^4) = 40x^3\)
- \(\frac{\partial}{\partial y} (3yz^6) = 3z^6\)
- \(\frac{\partial}{\partial z
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