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by Mary Bridget Kustusch (Fall 2011)

This field ($\vec{F}=Cz\hat{z}$) gets at many good issues such as the dot product in the definition of flux, but one that it doesn't address is the cross product nature of $\vec{da}$. To address this, it might be helpful to have a field that has both $\hat{z}$ and $\hat{r}$ components. Then, both terms from $\vec{da}$ have to be kept. If you are working in spherical components, $\vec{da}$ only has a $\hat{\theta}$ component and the difficulty there is to related $\hat{\theta}$ to the $\hat{z}$ in your field. An $\hat{r}$ component in $\vec{F}$ would disappear when dotted with $\hat{\theta}$.