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 such that . This means

Integrate both sides of the first PDE with respect to :

Differentating both sides with respect to gives

so that . A potential function exists, so the fundamental theorem does apply.

Green's theorem also applies because is a simple and smooth curve.

b. Now with (and I'm guessing as to what is supposed to be)

we want to find such that

Same procedure as in (a): integrating the first PDE wrt gives

Differentiating wrt gives

so that , which is undefined whenever , 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 as I did for , i.e.

Now

which is undefined at the point (0, 0). Again, both the fundamental theorem and Green's theorem apply.