Question

(a) Show that each of the vector fields F⃗ =4yi⃗ +4xj⃗ , G⃗ =3yx2+y2i⃗ +−3xx2+y2j⃗ , and H⃗ =2xx2+y2√i⃗ +2yx2+y2√j⃗ are gradient vector fields on some domain (not necessarily the whole plane) by finding a potential function for each. For F⃗ , a potential function is f(x,y)= 4xy For G⃗ , a potential function is g(x,y)= 4arctan(x/y) For H⃗ , a potential function is h(x,y)= (b) Find the line integrals of F⃗ ,G⃗ ,H⃗ around the curve C given to be the unit circle in the xy-plane, centered at the origin, and traversed counterclockwise. ∫CF⃗ ⋅dr⃗ = 0 ∫CG⃗ ⋅dr⃗ = ∫CH⃗ ⋅dr⃗ = 0 (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

Answer 1

Vector fields F, G, and H are shown to be gradient vector fields by finding their potential functions. The line integrals of F and H around the unit circle are zero, while the line integral of G requires direct computation or application of Green's Theorem. Green's Theorem is applicable only to G and H due to the restrictions on their domains.

• To demonstrate that the given vector fields F, G, and H are gradient vector fields on some domain, we can show that each vector field is the gradient of its potential function.
• For F, since f(x, y) = 4xy, the gradient ∇f = ∂f/∂x i + ∂f/∂y j will give us 4yi + 4xj, matching our original vector field F.
• Thus, F is indeed a gradient vector field.
• For G, the given potential function g(x, y) = 4arctan(x/y), the partial derivatives ∂g/∂x and ∂g/∂y will yield the vector field components of G, affirming that G is a gradient vector field.
• It should be noted that the domain for G excludes the y-axis where y = 0 to avoid undefined expressions in the arctangent function and division by zero.
• Similarly, for H, with the potential function h(x, y), the gradient will provide the vector field which confirms H as a gradient vector field.
• The domain for H must exclude the origin where x = 0 and y = 0 to avoid division by zero.
• The line integrals of F, G, and H around the curve C, which is the unit circle centered at the origin and traversed counterclockwise, can be computed using the fundamental theorem for line integrals since these are gradient vector fields on simply connected domains.
• For F and H, the line integrals are zero since these are conservative fields, but for G, we must calculate the integral directly or use Green's Theorem where applicable.
• Green's Theorem may be used to calculate the line integrals for G and H, but not for F, since F's potential function is not defined at the origin, a point enclosed by the curve C.
• Thus, the domain is not simply connected for F, which is a requirement for Green's Theorem to be applicable.

Answer 2

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Explanation:

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