cmpute the flux integral &F.nas where F=-4x yù+ 4 and CW +y2-1 oriented anti. the circle x? dock wLse.
cmpute the flux integral &F.nas where F=-4x yù+ 4 and CW +y2-1 oriented anti. the circle x? dock wLse.
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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I do not understand how to find the flux integral or what the symbol means.
![**Problem Statement:**
Compute the flux integral ∫C **F** · d**r**, where **F** = -4x **i** + 4y **j** and C is the circle x² + y² = 1 oriented anti-clockwise.
**Solution:**
1. Parameterize the circle x² + y² = 1 using:
- x = cos(t)
- y = sin(t)
- 0 ≤ t ≤ 2π
2. The vector field **F** becomes:
\[
\begin{align*}
F & = -4(\cos(t)) \, \mathbf{i} + 4(\sin(t)) \, \mathbf{j} \\
& = -4\cos(t) \, \mathbf{i} + 4\sin(t) \, \mathbf{j}
\end{align*}
\]
3. The differential arc length along C is given by:
\[
\begin{align*}
d\mathbf{r} & = \frac{d}{dt}(\cos(t), \sin(t)) \, dt \\
& = (-\sin(t) \, \mathbf{i} + \cos(t) \, \mathbf{j}) \, dt
\end{align*}
\]
4. Compute **F** · d**r**:
\[
\begin{align*}
\mathbf{F} \cdot d\mathbf{r} & = (-4\cos(t), 4\sin(t)) \cdot (-\sin(t), \cos(t)) \, dt \\
& = (4\cos(t)\sin(t) + 4\sin(t)\cos(t)) \, dt \\
& = 2\sin(2t) \, dt
\end{align*}
\]
5. Evaluate the integral:
\[
\begin{align*}
\int_0^{2\pi} 2\sin(2t) \, dt & = \left[-\cos(2t)\right]_0^{2\pi} \\
& = -\cos(4\pi) + \cos(0) \\
& = -1 + 1 \\
& = 0
\end{](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6e95c083-16cc-43f8-a531-1de0b0a0845b%2Fb35e7118-0e93-4684-8e8f-9b87e6cc7edb%2Fzzf177_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Compute the flux integral ∫C **F** · d**r**, where **F** = -4x **i** + 4y **j** and C is the circle x² + y² = 1 oriented anti-clockwise.
**Solution:**
1. Parameterize the circle x² + y² = 1 using:
- x = cos(t)
- y = sin(t)
- 0 ≤ t ≤ 2π
2. The vector field **F** becomes:
\[
\begin{align*}
F & = -4(\cos(t)) \, \mathbf{i} + 4(\sin(t)) \, \mathbf{j} \\
& = -4\cos(t) \, \mathbf{i} + 4\sin(t) \, \mathbf{j}
\end{align*}
\]
3. The differential arc length along C is given by:
\[
\begin{align*}
d\mathbf{r} & = \frac{d}{dt}(\cos(t), \sin(t)) \, dt \\
& = (-\sin(t) \, \mathbf{i} + \cos(t) \, \mathbf{j}) \, dt
\end{align*}
\]
4. Compute **F** · d**r**:
\[
\begin{align*}
\mathbf{F} \cdot d\mathbf{r} & = (-4\cos(t), 4\sin(t)) \cdot (-\sin(t), \cos(t)) \, dt \\
& = (4\cos(t)\sin(t) + 4\sin(t)\cos(t)) \, dt \\
& = 2\sin(2t) \, dt
\end{align*}
\]
5. Evaluate the integral:
\[
\begin{align*}
\int_0^{2\pi} 2\sin(2t) \, dt & = \left[-\cos(2t)\right]_0^{2\pi} \\
& = -\cos(4\pi) + \cos(0) \\
& = -1 + 1 \\
& = 0
\end{
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