Chapter 14 - Vector Calculus - 14.3 Conservative Vector Fields - 14.3 Exercises - Page 1086: 58

$-xy$

For a vector field to be Conservative, $\dfrac{\partial F_1}{\partial y}=\dfrac{\partial F_2}{\partial x}$ We are given that $F(x,y)=(-y, \ -x)$ Also, the curve is $r_1(t)= (t, 0) \implies r_1'(t) = (1, 0)$ and $r_2(t)= (x, t) \implies r_2'(t) = (0, 1)$ Therefore, the integral is: $\oint F \ dr=\int_0^{x} [F[r_1(t)] r_1'(t) \ dt +\int_0^{y} F[r_2(t)] r_2'(t)] \ dt\\=\int_0^{x} (0, -t) \cdot (1, 0) +\int_0^y (-t, -x) \cdot (0, 1) \ dt \\=-xy $ Thus, $\phi (x, y) =-xy$