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Let F₁ (x, y, z) = (y, x-z²+2, -2yz) and F₂(x, y, z) = (x − z², -y +2, 2yz). (a) Find the curl of each vector field. (b) Which of these two vector fields is conservative? Find a potential function for the one that is. (c) Let F be either F₁ or F2 (whichever you want), and let C denote the curve shown in the figure which starts at the point (0, 1, 0), stays above the circle x² + y² = 1 and spirals upwards in a clockwise direction to the point (0, 1, 1). Find the exact value of the line integral: Jo F. dr.

Let F₁ (x, y, z) = (y, x-z²+2, -2yz) and F₂(x, y, z) = (x − z², -y +2, 2yz). (a) Find the curl of each vector field. (b) Which of these two vector fields is conservative? Find a potential function for the one that is. (c) Let F be either F₁ or F2 (whichever you want), and let C denote the curve shown in the figure which starts at the point (0, 1, 0), stays above the circle x² + y² = 1 and spirals upwards in a clockwise direction to the point (0, 1, 1). Find the exact value of the line integral: Jo F. dr.

Advanced Engineering Mathematics
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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Let F₁ (x, y, z) = (y, x − z² +2,−2yz) and F₂ (x, y, z) = (x − z², −y + 2, 2yz). (a) Find the curl of each vector field. (b) Which of these two vector fields is conservative? Find a potential function for the one that is. (c) Let F be either F₁ or F2 (whichever you want), and let C denote the curve shown in the figure which starts at the point (0, 1,0), stays above the circle x² + y² = 1 and spirals upwards in a clockwise direction to the point (0,1,1). Jo Find the exact value of the line integral: F. dr.
Transcribed Image Text:Let F₁ (x, y, z) = (y, x − z² +2,−2yz) and F₂ (x, y, z) = (x − z², −y + 2, 2yz). (a) Find the curl of each vector field. (b) Which of these two vector fields is conservative? Find a potential function for the one that is. (c) Let F be either F₁ or F2 (whichever you want), and let C denote the curve shown in the figure which starts at the point (0, 1,0), stays above the circle x² + y² = 1 and spirals upwards in a clockwise direction to the point (0,1,1). Jo Find the exact value of the line integral: F. dr.
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