## Problem on qualitative evaluation of a line integral

Without calculation or application of any theorems, determine if $\int_C \mathbf{F}\cdot d\mathbf{r}$ is positive, negative, or zero. $\mathbf{F}$ is the vector field pictured below. $C$ is the circle traced out as indicated.
• ## Solution

Recall that
Recall that $d\mathbf{r}$ corresponds to an very small tangent vector along the path C.
Recall that
Along the bottom-right semicircle, $\mathbf{F}\cdot d\mathbf{r}$ is positive because $d\mathbf{r}$ points along $\mathbf{F}$.
Similarly, $\mathbf{F}\cdot d\mathbf{r}$ is negative along the top-left semicircle because $d\mathbf{r}$ points against $\mathbf{F}$.
It appears that $\mathbf{F}\cdot d\mathbf{r}$ takes on positive and negative values equally, so we suspect that $\int_C \mathbf{F}\cdot d\mathbf{r} = 0$
In order to verify that the line integral is zero, we break it into many pieces and see if we can see some cancellation with a piece on the opposite side of the circle. We draw two opposing pieces.
Observing that the vector $d\mathbf{r}$ on the bottom is of the same magnitude but opposite direction as the $d\mathbf{r}$ vector on the top, we can see that the contribution of $\mathbf{F}\cdot d\mathbf{r}$ from the bottom exactly cancels the contribution from the top.
Because the line integral is the sum of all segments forming the circle, and because each segment cancels the value of its opposite segment, $$\int_C \mathbf{F}\cdot d\mathbf{r} = 0$$