- #1

Granger

- 168

- 7

## Homework Statement

Compute the work of the vector field $$H: \mathbb{R^2} \setminus{(0,0}) \to \mathbb{R}$$

$$H(x,y)=\bigg(y^2-\frac{y}{x^2+y^2},1+2xy+\frac{x}{x^2+y^2}\bigg)$$

in the path $$g(t) = (1-t^2, t^2+t-1)$ with $t\in[-1,1]$$

## Homework Equations

3. The Attempt at a Solution [/B]

So first I considered my vector field as a sum of 2 vector fields: $$H = F + G$$

$$F(x,y)=\bigg(y^2,1+2xy\bigg)$$

$$G(x,y)=\bigg(-\frac{y}{x^2+y^2},\frac{x}{x^2+y^2}\bigg)$$

The vector field $$F$$ is conservative with one of many potentials $$A(x,y) = y^2x+y$$

Then I worked out the work using the definition and fundamental theorem of calculus obtaining the value 2.

So no problems at this point.

But $$G$$ is not a conservative vector field (it doesn't have a potential, even though it's a closed field). How should I proceed? I tried the definition but we get to a very complicated integral... The path isn't closed so we can't apply Green's theorem... What should I do?