Find the flux of the constant vector field =-i-2j+2k through a square plate of area 9 in the xy-plane oriented in the positive z-direction. flux =
Find the flux of the constant vector field =-i-2j+2k through a square plate of area 9 in the xy-plane oriented in the positive z-direction. flux =
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...
Related questions
Question
how do i solve the attached calculus problem?
![**Problem Statement:**
Find the flux of the constant vector field \( \vec{v} = -\vec{i} - 2\vec{j} + 2\vec{k} \) through a square plate of area 9 in the \( xy \)-plane oriented in the positive \( z \)-direction.
**Solution:**
To find the flux, we use the surface integral of the vector field across the given area. Since the area is in the \( xy \)-plane, the normal vector \( \vec{n} \) to the surface will be \( \vec{k} \).
Given:
- Vector Field: \( \vec{v} = -\vec{i} - 2\vec{j} + 2\vec{k} \)
- Area, \( A = 9 \)
- Normal Vector, \( \vec{n} = \vec{k} \)
The formula for flux \( \Phi \) is:
\[
\Phi = \int \int_S \vec{v} \cdot \vec{n} \, dS
\]
Since the vector field and normal vector are constants, and the integration is over a constant area, this simplifies to:
\[
\Phi = \vec{v} \cdot \vec{n} \cdot A
\]
Substituting the values:
\[
\vec{v} \cdot \vec{n} = (-\vec{i} - 2\vec{j} + 2\vec{k}) \cdot \vec{k} = 2
\]
Therefore, the flux is:
\[
\Phi = 2 \times 9 = 18
\]
Thus, the flux through the square plate is \(\boxed{18}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb3cfeff4-8ba6-46a8-98d4-804b4f4f620a%2F2fef3f18-5575-4065-8b2e-c58678723eb2%2F95nbmk4_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the flux of the constant vector field \( \vec{v} = -\vec{i} - 2\vec{j} + 2\vec{k} \) through a square plate of area 9 in the \( xy \)-plane oriented in the positive \( z \)-direction.
**Solution:**
To find the flux, we use the surface integral of the vector field across the given area. Since the area is in the \( xy \)-plane, the normal vector \( \vec{n} \) to the surface will be \( \vec{k} \).
Given:
- Vector Field: \( \vec{v} = -\vec{i} - 2\vec{j} + 2\vec{k} \)
- Area, \( A = 9 \)
- Normal Vector, \( \vec{n} = \vec{k} \)
The formula for flux \( \Phi \) is:
\[
\Phi = \int \int_S \vec{v} \cdot \vec{n} \, dS
\]
Since the vector field and normal vector are constants, and the integration is over a constant area, this simplifies to:
\[
\Phi = \vec{v} \cdot \vec{n} \cdot A
\]
Substituting the values:
\[
\vec{v} \cdot \vec{n} = (-\vec{i} - 2\vec{j} + 2\vec{k}) \cdot \vec{k} = 2
\]
Therefore, the flux is:
\[
\Phi = 2 \times 9 = 18
\]
Thus, the flux through the square plate is \(\boxed{18}\).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning