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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![### Finding the Outward Flux across the Domain D
Given the vector field:
\[
\mathbf{F}(x, y, z) = \frac{xi + yj + zk}{x^2 + y^2 + z^2}
\]
We need to find the outward flux of \(\mathbf{F}\) across the sphere defined by:
\[
x^2 + y^2 + z^2 = R^2
\]
### Components of the Vector Field
- \(\mathbf{F}(x, y, z)\) is a vector field with components:
\[
\mathbf{F}(x, y, z) = \left( \frac{x}{x^2 + y^2 + z^2} \right)\mathbf{i} + \left( \frac{y}{x^2 + y^2 + z^2} \right)\mathbf{j} + \left( \frac{z}{x^2 + y^2 + z^2} \right)\mathbf{k}
\]
### Sphere Equation
- The sphere has the equation:
\[
x^2 + y^2 + z^2 = R^2
\]
### Objective
Calculate the outward flux of the vector field \(\mathbf{F}\) across the sphere. This involves evaluating the surface integral of the vector field over the surface of the sphere.
The integral to be evaluated is typically represented as:
\[
\iint_{S} \mathbf{F} \cdot \mathbf{dS}
\]
where \(\mathbf{dS}\) represents the differential surface area vector on the sphere.
This formula uses divergences and can sometimes be simplified using the divergence theorem. Make sure the integrals are computed carefully to reflect accurate flux values across the surface given by the sphere.
In conclusion, this mathematical problem is aimed at utilizing vector calculus principles to find flux across spherical domains, which has applications in fields like electromagnetism and fluid dynamics.
For further assistance and step-by-step solutions, check the detailed examples on our educational website.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee1f5d00-2b06-4d9b-996b-2e4c16434e95%2F8efd82a1-8d64-45ce-b16f-485f92f6440b%2F0rr10sn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding the Outward Flux across the Domain D
Given the vector field:
\[
\mathbf{F}(x, y, z) = \frac{xi + yj + zk}{x^2 + y^2 + z^2}
\]
We need to find the outward flux of \(\mathbf{F}\) across the sphere defined by:
\[
x^2 + y^2 + z^2 = R^2
\]
### Components of the Vector Field
- \(\mathbf{F}(x, y, z)\) is a vector field with components:
\[
\mathbf{F}(x, y, z) = \left( \frac{x}{x^2 + y^2 + z^2} \right)\mathbf{i} + \left( \frac{y}{x^2 + y^2 + z^2} \right)\mathbf{j} + \left( \frac{z}{x^2 + y^2 + z^2} \right)\mathbf{k}
\]
### Sphere Equation
- The sphere has the equation:
\[
x^2 + y^2 + z^2 = R^2
\]
### Objective
Calculate the outward flux of the vector field \(\mathbf{F}\) across the sphere. This involves evaluating the surface integral of the vector field over the surface of the sphere.
The integral to be evaluated is typically represented as:
\[
\iint_{S} \mathbf{F} \cdot \mathbf{dS}
\]
where \(\mathbf{dS}\) represents the differential surface area vector on the sphere.
This formula uses divergences and can sometimes be simplified using the divergence theorem. Make sure the integrals are computed carefully to reflect accurate flux values across the surface given by the sphere.
In conclusion, this mathematical problem is aimed at utilizing vector calculus principles to find flux across spherical domains, which has applications in fields like electromagnetism and fluid dynamics.
For further assistance and step-by-step solutions, check the detailed examples on our educational website.
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