Use Green's Theorem to find the counterclockwise circulation and outward flux for the field F and curve C. 32 5 F = 5x³y²i+ 2 yj The outward flux is 3645 7 (Type an integer or a simplified fraction.) The counterclockwise circulation is (Type an integer or a simplified fraction.) (...) (0,0) y=x (3,3) C y = x - 2x X
Use Green's Theorem to find the counterclockwise circulation and outward flux for the field F and curve C. 32 5 F = 5x³y²i+ 2 yj The outward flux is 3645 7 (Type an integer or a simplified fraction.) The counterclockwise circulation is (Type an integer or a simplified fraction.) (...) (0,0) y=x (3,3) C y = x - 2x X
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 Green's Theorem for Counterclockwise Circulation and Outward Flux**
### Problem Statement:
Utilize Green's Theorem to determine the counterclockwise circulation and outward flux for the vector field \( \mathbf{F} \) and the curve \( C \).
\[ \mathbf{F} = 5x^3y^2 \mathbf{i} + \frac{5}{2} x^4 y \mathbf{j} \]
### Diagram Explanation:
On the right side, there is a Cartesian coordinate system with two curves forming a closed loop denoted as \( C \).
- The curve \( y = x \) extends from the point \( (0,0) \) to \( (3,3) \).
- The curve \( y = x^2 - 2x \) extends from the point \( (3,3) \) back to \( (0,0) \).
### Calculation Details:
**1. Outward Flux:**
\[ \text{The outward flux is} \; \frac{3645}{7} \; . \]
(Type an integer or a simplified fraction.)
**2. Counterclockwise Circulation:**
\[ \text{The counterclockwise circulation is} \; \_\_\_\_\; \]
(Type an integer or a simplified fraction.)
---
Green's Theorem is a powerful tool in vector calculus, relating a line integral around a simple closed curve \( C \) to a double integral over the plane region \( D \) bounded by \( C \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0c6d6b24-4b7a-4949-954a-bc8418c314a2%2F99ea3ac8-fe5c-4d15-9b3d-be2960f33a3d%2Fs5bzo7_processed.png&w=3840&q=75)
Transcribed Image Text:**Using Green's Theorem for Counterclockwise Circulation and Outward Flux**
### Problem Statement:
Utilize Green's Theorem to determine the counterclockwise circulation and outward flux for the vector field \( \mathbf{F} \) and the curve \( C \).
\[ \mathbf{F} = 5x^3y^2 \mathbf{i} + \frac{5}{2} x^4 y \mathbf{j} \]
### Diagram Explanation:
On the right side, there is a Cartesian coordinate system with two curves forming a closed loop denoted as \( C \).
- The curve \( y = x \) extends from the point \( (0,0) \) to \( (3,3) \).
- The curve \( y = x^2 - 2x \) extends from the point \( (3,3) \) back to \( (0,0) \).
### Calculation Details:
**1. Outward Flux:**
\[ \text{The outward flux is} \; \frac{3645}{7} \; . \]
(Type an integer or a simplified fraction.)
**2. Counterclockwise Circulation:**
\[ \text{The counterclockwise circulation is} \; \_\_\_\_\; \]
(Type an integer or a simplified fraction.)
---
Green's Theorem is a powerful tool in vector calculus, relating a line integral around a simple closed curve \( C \) to a double integral over the plane region \( D \) bounded by \( C \).
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