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(a) Show that each of the vector fields F = 4yi + 4x j, Ğ = 24² +247 3₁ and H = √2+1+34 are gradient vector fields on some domain (not necessarily the whole plane) by finding a potential function for each. For F, a potential function is f(x, y) = For G, a potential function is g(x, y) = For H, a potential function is h(x, y) = 9. (b) Find the line integrals of F, G, H around the curve C given to be the unit circle in the xy-plane, centered at the origin, and traversed counterclockwise. ScF.dr = ScĠ. dr = ScH. dr = (c) For which of the three vector fields can Green's Theorem be used to calculate the line integral in part (b)? It may be used for?

(a) Show that each of the vector fields F = 4yi + 4x j, Ğ = 24² +247 3₁ and H = √2+1+34 are gradient vector fields on some domain (not necessarily the whole plane) by finding a potential function for each. For F, a potential function is f(x, y) = For G, a potential function is g(x, y) = For H, a potential function is h(x, y) = 9. (b) Find the line integrals of F, G, H around the curve C given to be the unit circle in the xy-plane, centered at the origin, and traversed counterclockwise. ScF.dr = ScĠ. dr = ScH. dr = (c) For which of the three vector fields can Green's Theorem be used to calculate the line integral in part (b)? It may be used for?

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter8: Applications Of Trigonometry
Section8.4: The Dot Product
Problem 12E
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(a) Show that each of the vector fields F = 4yi + 4x j‚ Ġ = 5i + 5 j, and Ĥ = √x²+y² finding a potential function for each. For F, a potential function is f(x, y) = For G, a potential function is g(x, y): = + are gradient vector fields on some domain (not necessarily the whole plane) by For H, a potential function is h(x, y) (b) Find the line integrals of F, G, H around the curve C given to be the unit circle in the xy-plane, centered at the origin, and traversed counterclockwise. ScF. dr = ScĞ.dr = ScH. dr = (c) For which of the three vector fields can Green's Theorem be used to calculate the line integral in part (b)? It may be used for?
Transcribed Image Text:(a) Show that each of the vector fields F = 4yi + 4x j‚ Ġ = 5i + 5 j, and Ĥ = √x²+y² finding a potential function for each. For F, a potential function is f(x, y) = For G, a potential function is g(x, y): = + are gradient vector fields on some domain (not necessarily the whole plane) by For H, a potential function is h(x, y) (b) Find the line integrals of F, G, H around the curve C given to be the unit circle in the xy-plane, centered at the origin, and traversed counterclockwise. ScF. dr = ScĞ.dr = ScH. dr = (c) For which of the three vector fields can Green's Theorem be used to calculate the line integral in part (b)? It may be used for?
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