Consider the vector field F = (x¹y³, x³y¹) . O The vector field is not conservative The vector field is conservative, and the potential function for Fis (use K for the constant) (x, y) LE If is conservative, use ☀(x, y) to evaluate along a piecewise smooth curve (C) from (4,5) to (-1,-3). Please show an exact answer or an approximation to at least 4 significant digits. Li F.dr F.dr
Consider the vector field F = (x¹y³, x³y¹) . O The vector field is not conservative The vector field is conservative, and the potential function for Fis (use K for the constant) (x, y) LE If is conservative, use ☀(x, y) to evaluate along a piecewise smooth curve (C) from (4,5) to (-1,-3). Please show an exact answer or an approximation to at least 4 significant digits. Li F.dr 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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Question
5.4-5
![Consider the vector field \( \vec{F} = \langle x^4 y^3, x^3 y^4 \rangle \).
- [ ] The vector field is not conservative.
- [ ] The vector field is conservative, and the potential function for \( \vec{F} \) is (use \( K \) for the constant)
\[
\phi(x, y) = \boxed{\phantom{answer}}
\]
If \( \vec{F} \) is conservative, use \( \phi(x, y) \) to evaluate \( \int_{C} \vec{F} \cdot d\vec{r} \)
along a piecewise smooth curve \( C \) from \( (4,5) \) to \((-1,-3)\). Please show an exact answer or an approximation to at least 4 significant digits.
\[
\int_{C} \vec{F} \cdot d\vec{r} = \boxed{\phantom{answer}}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4e39a1d0-143c-41f0-ae59-e967e0535bad%2F937b77ca-bc18-428a-91bf-ea285b8534b5%2Fv133cm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the vector field \( \vec{F} = \langle x^4 y^3, x^3 y^4 \rangle \).
- [ ] The vector field is not conservative.
- [ ] The vector field is conservative, and the potential function for \( \vec{F} \) is (use \( K \) for the constant)
\[
\phi(x, y) = \boxed{\phantom{answer}}
\]
If \( \vec{F} \) is conservative, use \( \phi(x, y) \) to evaluate \( \int_{C} \vec{F} \cdot d\vec{r} \)
along a piecewise smooth curve \( C \) from \( (4,5) \) to \((-1,-3)\). Please show an exact answer or an approximation to at least 4 significant digits.
\[
\int_{C} \vec{F} \cdot d\vec{r} = \boxed{\phantom{answer}}
\]
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