For each of the following vector fields F, decide whether it is conservative or not by computing the appropriate first order partial derivatives. Type in a potential function f (that is, Vƒ = F) with ƒ(0,0) = 0. If it is not conservative, type N. A. F (x, y) = (−16x + 8y)i + (8x + 12y) j f (x, y) = 3. F(x, y) = -8yi - 7xj f (x, y) = c. F (x, y) = (−8 sin y)i + (16y − 8x cos y) j f (x, y) =
For each of the following vector fields F, decide whether it is conservative or not by computing the appropriate first order partial derivatives. Type in a potential function f (that is, Vƒ = F) with ƒ(0,0) = 0. If it is not conservative, type N. A. F (x, y) = (−16x + 8y)i + (8x + 12y) j f (x, y) = 3. F(x, y) = -8yi - 7xj f (x, y) = c. F (x, y) = (−8 sin y)i + (16y − 8x cos y) j f (x, y) =
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
![For each of the following vector fields \( \mathbf{F} \), decide whether it is conservative or not by computing the appropriate first order partial derivatives. Type in a potential function \( f \) (that is, \( \nabla f = \mathbf{F} \)) with \( f(0, 0) = 0 \). If it is not conservative, type N.
A. \( \mathbf{F}(x, y) = (-16x + 8y) \mathbf{i} + (8x + 12y) \mathbf{j} \)
\( f(x, y) = \) [Text box for input]
B. \( \mathbf{F}(x, y) = -8y \mathbf{i} - 7x \mathbf{j} \)
\( f(x, y) = \) [Text box for input]
C. \( \mathbf{F}(x, y) = (-8 \sin y) \mathbf{i} + (16y - 8x \cos y) \mathbf{j} \)
\( f(x, y) = \) [Text box for input]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18fe6d10-65c6-4dea-a463-83c487832ab0%2Fc23ff30c-bba8-4cb9-ba4b-fafec4c9e7b4%2Febq9jo_processed.png&w=3840&q=75)
Transcribed Image Text:For each of the following vector fields \( \mathbf{F} \), decide whether it is conservative or not by computing the appropriate first order partial derivatives. Type in a potential function \( f \) (that is, \( \nabla f = \mathbf{F} \)) with \( f(0, 0) = 0 \). If it is not conservative, type N.
A. \( \mathbf{F}(x, y) = (-16x + 8y) \mathbf{i} + (8x + 12y) \mathbf{j} \)
\( f(x, y) = \) [Text box for input]
B. \( \mathbf{F}(x, y) = -8y \mathbf{i} - 7x \mathbf{j} \)
\( f(x, y) = \) [Text box for input]
C. \( \mathbf{F}(x, y) = (-8 \sin y) \mathbf{i} + (16y - 8x \cos y) \mathbf{j} \)
\( f(x, y) = \) [Text box for input]
Expert Solution
Step 1
The given vector fields are:
A.
B.
C.
We have to check whether the given fields are conservative or not. Then find the potential function.
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