What is the work done along a quarter circle in a particular vector field?
To better understand different types of line integrals, I set up the problem below, and request verification:
Let $F$ be a vector field such that $F(x,y) = \begin{bmatrix}x \\ 1\end{bmatrix}$.
Let $P$ be the path from $(0,1)$ to $(1,0)$ along the unit circle.
What is $\int_P F(x,y) \cdot d \ell$ (that is, the work done along $P$)?
Solution
Parameterize $P$ by $s$, the total distance traveled. Then $$\ell(s) = \begin{bmatrix}\sin s \\ \cos s\end{bmatrix} \\ \frac {d \ell}{d s} = \begin{bmatrix}\cos s \\ -\sin s\end{bmatrix} \\ F(\ell(s)) = \begin{bmatrix}\sin s \\ 1 \end{bmatrix}$$ and the path goes from $s = 0$ to $s = \frac \pi 2$.
So $$\int_P F(x,y) \cdot d \ell = \int_0^{\pi / 2}F(\ell(s)) \cdot \frac {d \ell}{d s} ds = \int_0^{\pi / 2} [\sin s \cos s - \sin s] ds = - \frac 1 2.$$