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F₁ = (e + sin(y), xe" - 2y)

F₁ = (e + sin(y), xe" - 2y)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.6: Additional Trigonometric Graphs
Problem 78E
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7. (Note: The three parts of this question are unrelated. Each one is about different vector fields.) (a) For each of the three vector fields below, first determine whether it is conservative and then find a potential function if it is: F₁ = (e + sin(y), xey – 2y) F₂ = (e³+ sin(x), xe² + 2y) 2 F3 = (xe" - sin(x), e² + 2y) (b) The vector field F = (7xy, e) is not conservative. Find the exact value of the integral of this vector field along the path C parametrized as r(t) = (t², t³) for −1 ≤t≤1. (c) The vector field F(x, y) = (-y sin(xy), -x sin(xy)) is the gradient vector field of the function f(x, y) = cos(xy). (That is, F = Vf.) Use this fact to evaluate the line integral where C is parametrized by r(t) = (1+½, t²-2t+π) for 0 ≤ t ≤ 2.
Transcribed Image Text:7. (Note: The three parts of this question are unrelated. Each one is about different vector fields.) (a) For each of the three vector fields below, first determine whether it is conservative and then find a potential function if it is: F₁ = (e + sin(y), xey – 2y) F₂ = (e³+ sin(x), xe² + 2y) 2 F3 = (xe" - sin(x), e² + 2y) (b) The vector field F = (7xy, e) is not conservative. Find the exact value of the integral of this vector field along the path C parametrized as r(t) = (t², t³) for −1 ≤t≤1. (c) The vector field F(x, y) = (-y sin(xy), -x sin(xy)) is the gradient vector field of the function f(x, y) = cos(xy). (That is, F = Vf.) Use this fact to evaluate the line integral where C is parametrized by r(t) = (1+½, t²-2t+π) for 0 ≤ t ≤ 2.
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are F2 and F3 conservative? also what about parts B and C?

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