For each of the following vector fields F, decide whether it is conservative or not by computing curl F. Type in a potential function f (that is, Vƒ = F). If it is not conservative, type N. A. F (x, y) = (–10x − 1y)i + (−1x + 10y)j f (x, y) = B. F(x, y) = -5yi - 4xj f (x, y) = C. F (x, y, z) = −5xi − 4yj + k f (x, y, z) = D. F (x, y) = (–5 sin y) i + (−2y – 5x cos y) j f (x, y) = E. F(x, y, z) = -5x²i − 1y²j + 5z²k f (x, y, z) =
For each of the following vector fields F, decide whether it is conservative or not by computing curl F. Type in a potential function f (that is, Vƒ = F). If it is not conservative, type N. A. F (x, y) = (–10x − 1y)i + (−1x + 10y)j f (x, y) = B. F(x, y) = -5yi - 4xj f (x, y) = C. F (x, y, z) = −5xi − 4yj + k f (x, y, z) = D. F (x, y) = (–5 sin y) i + (−2y – 5x cos y) j f (x, y) = E. F(x, y, z) = -5x²i − 1y²j + 5z²k f (x, y, z) =
Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Task: Determine if Vector Fields are Conservative**
For each of the following vector fields \( \mathbf{F} \), decide whether it is conservative or not by computing \( \text{curl} \, \mathbf{F} \). Type in a potential function \( f \) (that is, \( \nabla f = \mathbf{F} \)). If it is not conservative, type N.
A. \( \mathbf{F}(x, y) = (-10x - 1y) \mathbf{i} + (-1x + 10y) \mathbf{j} \)
\( f(x, y) = \) [ ]
B. \( \mathbf{F}(x, y) = -5y \mathbf{i} - 4x \mathbf{j} \)
\( f(x, y) = \) [ ]
C. \( \mathbf{F}(x, y, z) = -5x \mathbf{i} - 4y \mathbf{j} + \mathbf{k} \)
\( f(x, y, z) = \) [ ]
D. \( \mathbf{F}(x, y) = (-5 \sin y) \mathbf{i} + (-2y - 5x \cos y) \mathbf{j} \)
\( f(x, y) = \) [ ]
E. \( \mathbf{F}(x, y, z) = -5x^2 \mathbf{i} - 1y^2 \mathbf{j} + 5z^2 \mathbf{k} \)
\( f(x, y, z) = \) [ ]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18fe6d10-65c6-4dea-a463-83c487832ab0%2F2b071aab-9aaf-4250-82fd-2c0b105284f1%2Fox0rp3j_processed.png&w=3840&q=75)
Transcribed Image Text:**Task: Determine if Vector Fields are Conservative**
For each of the following vector fields \( \mathbf{F} \), decide whether it is conservative or not by computing \( \text{curl} \, \mathbf{F} \). Type in a potential function \( f \) (that is, \( \nabla f = \mathbf{F} \)). If it is not conservative, type N.
A. \( \mathbf{F}(x, y) = (-10x - 1y) \mathbf{i} + (-1x + 10y) \mathbf{j} \)
\( f(x, y) = \) [ ]
B. \( \mathbf{F}(x, y) = -5y \mathbf{i} - 4x \mathbf{j} \)
\( f(x, y) = \) [ ]
C. \( \mathbf{F}(x, y, z) = -5x \mathbf{i} - 4y \mathbf{j} + \mathbf{k} \)
\( f(x, y, z) = \) [ ]
D. \( \mathbf{F}(x, y) = (-5 \sin y) \mathbf{i} + (-2y - 5x \cos y) \mathbf{j} \)
\( f(x, y) = \) [ ]
E. \( \mathbf{F}(x, y, z) = -5x^2 \mathbf{i} - 1y^2 \mathbf{j} + 5z^2 \mathbf{k} \)
\( f(x, y, z) = \) [ ]
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