Use the Divergence Theorem to find the outward flux of F = (7x° + 12xy) i+ (3y° +2esin z) j+ (7z° +2ecos z) k across the boundary of the region D: the solid region between the spheres x + y² +z? = 1 and x +y? +z? = 2. F3D The outward flux of F = (7x + 12xy²) i + (3y° +2e'sin z) j+ (7z + 2ecos z) k is (Type an exact anwer, using as needed.)
Use the Divergence Theorem to find the outward flux of F = (7x° + 12xy) i+ (3y° +2esin z) j+ (7z° +2ecos z) k across the boundary of the region D: the solid region between the spheres x + y² +z? = 1 and x +y? +z? = 2. F3D The outward flux of F = (7x + 12xy²) i + (3y° +2e'sin z) j+ (7z + 2ecos z) k is (Type an exact anwer, using as needed.)
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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![### Divergence Theorem Application
Use the Divergence Theorem to find the outward flux of **F** given by:
\[ \mathbf{F} = \left( 7x^3 + 12xy^2 \right) \mathbf{i} + \left( 3y^3 + 2e^y \sin z \right) \mathbf{j} + \left( 7z^3 + 2e^y \cos z \right) \mathbf{k} \]
across the boundary of the region \( D \): the solid region between the spheres \( x^2 + y^2 + z^2 = 1 \) and \( x^2 + y^2 + z^2 = 2 \).
#### Problem Statement:
The outward flux of:
\[ \mathbf{F} = \left( 7x^3 + 12xy^2 \right) \mathbf{i} + \left( 3y^3 + 2e^y \sin z \right) \mathbf{j} + \left( 7z^3 + 2e^y \cos z \right) \mathbf{k} \]
is \( \boxed{} \). (Type an exact answer, using \(\pi\) as needed.)
### Steps to Solve:
1. **Calculate the divergence of** \( \mathbf{F} \):
\[ \nabla \cdot \mathbf{F} = \frac{\partial}{\partial x} (7x^3 + 12xy^2) + \frac{\partial}{\partial y} (3y^3 + 2e^y \sin z) + \frac{\partial}{\partial z} (7z^3 + 2e^y \cos z) \]
2. **Integrate the divergence over the volume \( D \)**:
\[ \text{Flux} = \iiint_D (\nabla \cdot \mathbf{F}) \, dV \]
3. **Use spherical coordinates transformation for the bounds of integration**, considering the given spheres \( x^2 + y^2 + z^2 = 1 \) and \( x^2 + y^2 + z^2 = 2 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1bedffc2-fb9b-46af-a153-6d9ee06b6f91%2Fbd7bb45f-6991-4eaf-b55d-d5e622a2624d%2Fayqpzpa_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Divergence Theorem Application
Use the Divergence Theorem to find the outward flux of **F** given by:
\[ \mathbf{F} = \left( 7x^3 + 12xy^2 \right) \mathbf{i} + \left( 3y^3 + 2e^y \sin z \right) \mathbf{j} + \left( 7z^3 + 2e^y \cos z \right) \mathbf{k} \]
across the boundary of the region \( D \): the solid region between the spheres \( x^2 + y^2 + z^2 = 1 \) and \( x^2 + y^2 + z^2 = 2 \).
#### Problem Statement:
The outward flux of:
\[ \mathbf{F} = \left( 7x^3 + 12xy^2 \right) \mathbf{i} + \left( 3y^3 + 2e^y \sin z \right) \mathbf{j} + \left( 7z^3 + 2e^y \cos z \right) \mathbf{k} \]
is \( \boxed{} \). (Type an exact answer, using \(\pi\) as needed.)
### Steps to Solve:
1. **Calculate the divergence of** \( \mathbf{F} \):
\[ \nabla \cdot \mathbf{F} = \frac{\partial}{\partial x} (7x^3 + 12xy^2) + \frac{\partial}{\partial y} (3y^3 + 2e^y \sin z) + \frac{\partial}{\partial z} (7z^3 + 2e^y \cos z) \]
2. **Integrate the divergence over the volume \( D \)**:
\[ \text{Flux} = \iiint_D (\nabla \cdot \mathbf{F}) \, dV \]
3. **Use spherical coordinates transformation for the bounds of integration**, considering the given spheres \( x^2 + y^2 + z^2 = 1 \) and \( x^2 + y^2 + z^2 = 2 \).
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