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F₁ = (e + sin(y), xe" - 2y)

F₁ = (e + sin(y), xe" - 2y)

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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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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