Find the line integral for the following vector field

Question

$\vec{G} = \big(\frac{x}{\sqrt{x^2+y^2-1}}, \frac{y}{\sqrt{x^2+y^2-1}}\big)$.
I need to calculate the line integral $\int_{\mathrm{CR}}\vec {G}\, d\vec{r}\, $ where $CR$ is a circle with the center in origin and the radius $1 \lt R.$ I know that $x^2+y^2 \gt 1,$ which means all points outside of the circle with radius $1$ and center in origin. How do you calculate the line integral of such a vector field?

Answer 1

You can establish that $\vec G$ is conservative, meaning there is some scalar function $g(x,y)$ such that $\nabla g(x,y)=\vec G$. In particular, you have

$$g(x,y)=\sqrt{x^2+y^2-1}+c$$

since

$$\begin{cases} \dfrac{\partial g}{\partial x}=\dfrac x{\sqrt{x^2+y^2-1}}\\[1ex] \dfrac{\partial g}{\partial y}=\dfrac y{\sqrt{x^2+y^2-1}} \end{cases}$$

where $c$ is an arbitrary constant. Then by the gradient theorem, the line integral over the given curve is $0$.