Shown in Figure 13.2.10 are two vector fields, \(\vF\) and \(\vG\) and four oriented curves, as labeled in the plots. For each of the line integrals below, determine if its value should be positive, negative, or zero. Do this by thinking about if the vector field is helping or hindering a particle moving along the oriented curve, rather than by doing calculations.
A vector field radiating from the origin with vectors getting longer as distance from the origin increases. There is an oriented line segment labeled \(C_1\) from \((-2,-2)\) to \((2,2)\) and an oriented line segment labeled \(C_2\) from \((2,-2)\) to \((0,-2)\text{.}\)
A vector field with all vectors parallel to the \(y\)-axis. Vectors get longer as distance from the \(y\)-axis increases. Vectors with \(x>0\) point in the positive \(y\)-direction, while vectors with \(x\lt 0\) point in the negative \(y\)-direction. The top half of the circle of radius \(2.5\) centered at the origin and oriented clockwise is labeled \(C_3\text{.}\) There is an oriented line segment labeled \(C_4\) from \((2,-2)\) to \((0,-2)\text{.}\)
(b)A plot of \(\vG\) with paths \(C_3\) and \(C_4\)
