Hey I want to solve this problem:
Let $F(x,y)=(cos(xy),g(x,y))$. Find a function $g(x,y)$ such that
$$ \int_{C} F=0$$
Where $C$ is the circle of radius $5$ and center $(1,-2)$.
So I tried to solve this by using the Green's Theorem, because by the "general" way to solve line integrals of vector fields results in a very complicated integral.
Applying Green's Theorem I need to find a $g(x,y)$ that makes zero the integral:
$$ \int_{-4}^{6} \int_{-a-2}^{a-2} \frac{\partial g}{\partial x}+ xsin(xy) \hspace{1mm} dy dx $$
Where $a= \sqrt{5^2-(x-1)^2}$
From here I said that one $g(x,y)$ can be
$$ \frac{\partial g}{\partial x}=- xsin(xy) $$
By integration:
$$ g(x,y)= -\frac{1}{y^2} \Big( -xycos(xy)+sin(xy)+C \Big) $$
But, I'm not 100% sure that this solution can work... because that solution only help with $y \neq 0$...
How would you solve it? Any suggestion?