Find fF.dr for F = 8yi - (sin y)j on the curve counterclockwise around the unit circle C starting at the point (1,0). Sc F. dr =
Find fF.dr for F = 8yi - (sin y)j on the curve counterclockwise around the unit circle C starting at the point (1,0). Sc F. dr =
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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how do i solve the attached calculus question?
![**Problem Statement:**
Find the line integral \( \int_C \mathbf{F} \cdot d\mathbf{r} \) for the vector field \( \mathbf{F} = 8y\mathbf{i} - (\sin y)\mathbf{j} \) on the curve \( C \), which is the unit circle traversed counterclockwise starting at the point \( (1, 0) \).
**Solution:**
The goal is to compute \( \int_C \mathbf{F} \cdot d\mathbf{r} \).
**Explanation:**
1. **Vector Field \( \mathbf{F} = 8y\mathbf{i} - (\sin y)\mathbf{j} \):**
- The vector field \( \mathbf{F} \) is composed of two components: \( 8y \) in the direction of \( \mathbf{i} \) and \( -(\sin y) \) in the direction of \( \mathbf{j} \).
2. **Curve \( C \):**
- Curve \( C \) is the unit circle, defined as \( x^2 + y^2 = 1 \).
- The traversal is counterclockwise, starting from the point \( (1, 0) \).
3. **Integral \( \int_C \mathbf{F} \cdot d\mathbf{r} \):**
- This integral assesses how the vector field \( \mathbf{F} \) behaves along the path \( C \).
Educational aspects to consider include understanding the computation of line integrals, the behavior of vector fields, and the significance of path direction and parameterization.
**Boxed Area:**
\[ \int_C \mathbf{F} \cdot d\mathbf{r} = \boxed{?} \]
---
*Note: Add calculations and results once solved, explaining each step and decision in the process.*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb3cfeff4-8ba6-46a8-98d4-804b4f4f620a%2Ff40a1e02-983b-4b0d-bc92-d5a657fb8209%2Fovi6mp5_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the line integral \( \int_C \mathbf{F} \cdot d\mathbf{r} \) for the vector field \( \mathbf{F} = 8y\mathbf{i} - (\sin y)\mathbf{j} \) on the curve \( C \), which is the unit circle traversed counterclockwise starting at the point \( (1, 0) \).
**Solution:**
The goal is to compute \( \int_C \mathbf{F} \cdot d\mathbf{r} \).
**Explanation:**
1. **Vector Field \( \mathbf{F} = 8y\mathbf{i} - (\sin y)\mathbf{j} \):**
- The vector field \( \mathbf{F} \) is composed of two components: \( 8y \) in the direction of \( \mathbf{i} \) and \( -(\sin y) \) in the direction of \( \mathbf{j} \).
2. **Curve \( C \):**
- Curve \( C \) is the unit circle, defined as \( x^2 + y^2 = 1 \).
- The traversal is counterclockwise, starting from the point \( (1, 0) \).
3. **Integral \( \int_C \mathbf{F} \cdot d\mathbf{r} \):**
- This integral assesses how the vector field \( \mathbf{F} \) behaves along the path \( C \).
Educational aspects to consider include understanding the computation of line integrals, the behavior of vector fields, and the significance of path direction and parameterization.
**Boxed Area:**
\[ \int_C \mathbf{F} \cdot d\mathbf{r} = \boxed{?} \]
---
*Note: Add calculations and results once solved, explaining each step and decision in the process.*
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