Example 2.3.7: Line integral along the z-axis

The path is the segment $C_2$ of the $z$-axis from the origin to $(0,0,2)$, with $d\ell =dz{a}_z$. Then $A\cdot d\ell =A_{z}dz=-z^{3}dz$, and

The negative sign means that along this path the field points, on average, against the direction of travel (indeed $A_z=-z^3\le 0$ for $z\ge 0$).