## Problem on qualitative evaluation of line integrals

Without calculation, 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 red circle traced out clockwise.
• ## Solution

Recall that
We begin by determining the sign of $\mathbf{F} \cdot d\mathbf{r}$ along $C$.
Recall that $d\mathbf{r}$ corresponds to an very small tangent vector along the path C.
Recall that
If the angle between two vectors is less than $\pi/2$, their dot product is positive.
Because $\mathbf{F}$ and $d\mathbf{r}$ are parallel, the angle between them is 0. Hence $\bfF\cdot d\bfr$ is everywhere on $C$.
Because the integrand is always positive, $\int_C \mathbf{F}\cdot d\mathbf{r}$ is postiive.