Let C be a circle of radius 9 centered at (0,0), traversed counterclockwise. Use this curve to answer the questions below. (a) Let F = y i + x j Find a potential function. f(x,y) = xy Does the Fundamental Theorem of Line Integrals apply to F · dr C ? Does Green's Theorem apply to F · dr C ? (b) Let G = y x2 + y2 i − x x2 + y2 j . Find a potential function. g(x,y) = Where is the potential function not defined? Does the Fundamental Theorem of Line Integrals apply to G · dr C ? Does Green's Theorem apply to G · dr C ? (c) Let H = x x2 + y2 i + y x2 + y2 j . Find a potential function. h(x,y) = Where is the potential function not defined? Does the Fundamental Theorem of Line Integrals apply to H · dr C ? Does Green's Theorem apply to H · dr C ? Submit Answer Save Progress

a. We're looking for a scalar function [tex]f(x,y)[/tex] such that [tex]\vec F(x,y)=\nabla f(x,y)[/tex]. This means[tex]\dfrac{\partial f}{\partial x}=y[/tex][tex]\dfrac{\partial f}{\partial y}=x[/tex]Integrate both sides of the first PDE with respect to [tex]x[/tex]:[tex]\displaystyle\int\frac{\partial f}{\partial x}\,\mathrm dx=\int y\,\mathrm dx\implies f(x,y)=xy+g(y)[/tex]Differentating both sides with respect to [tex]y[/tex] gives[tex]\dfrac{\partial f}{\partial y}=x=x+\dfrac{\mathrm dg}{\mathrm dy}\implies g(y)=C[/tex]so that [tex]\boxed{f(x,y)=xy+C}[/tex]. A potential function exists, so the fundamental theorem does apply.Green's theorem also applies because [tex]C[/tex] is a simple and smooth curve.b. Now with (and I'm guessing as to what [tex]\vec G[/tex] is supposed to be)[tex]\vec G(x,y)=\dfrac y{x^2+y^2}\,\vec\imath-\dfrac x{x^2+y^2}\,\vec\jmath[/tex]we want to find [tex]g[/tex] such that[tex]\dfrac{\partial g}{\partial x}=\dfrac y{x^2+y^2}[/tex][tex]\dfrac{\partial g}{\partial y}=-\dfrac x{x^2+y^2}[/tex]Same procedure as in (a): integrating the first PDE wrt [tex]x[/tex] gives[tex]g(x,y)=\tan^{-1}\dfrac xy+h(y)[/tex]Differentiating wrt [tex]y[/tex] gives[tex]-\dfrac x{x^2+y^2}=-\dfrac x{x^2+y^2}+\dfrac{\mathrm dh}{\mathrm dy}\implies h(y)=C[/tex]so that [tex]\boxed{g(x,y)=\tan^{-1}\dfrac xy+C}[/tex], which is undefined whenever [tex]y=0[/tex], and the fundamental theorem applies, and Green's theorem also applies for the same reason as in (a).c. Same as (b) with slight changes. Again, I'm assuming the same format for [tex]\vec H[/tex] as I did for [tex]\vec G[/tex], i.e.[tex]\vec H(x,y)=\dfrac x{x^2+y^2}\,\vec\imath+\dfrac y{x^2+y^2}\,\vec\jmath[/tex]Now[tex]\dfrac{\partial h}{\partial x}=\dfrac x{x^2+y^2}\implies h(x,y)=\dfrac12\ln(x^2+y^2)+i(y)[/tex][tex]\dfrac{\partial h}{\partial x}=\dfrac y{x^2+y^2}=\dfrac y{x^2+y^2}+\dfrac{\mathrm di}{\mathrm dy}\implies i(y)=C[/tex][tex]\implies\boxed{h(x,y)=\dfrac12\ln(x^2+y^2)+C}[/tex]which is undefined at the point (0, 0). Again, both the fundamental theorem and Green's theorem apply.