After finishing any leftovers from Friday, we'll calculate the flux of a vector function $\vec{F}$ across various surfaces (14.6).

In this drawing, the flux of $\vec{F}$ across the oriented surface element $d\vec{S}$ is represented by the volume of the gray boxes. You can see that the flux is largest when $\vec{F}$ and $d\vec{S}$ are parallel, but decreases as the angle between the vectors increases. In detail, the differential of flux is equal to the dot product between $\vec{F}$ and $d\vec{S}$ and the flux through an entire surface is $$\iint \vec{F} \cdot d\vec{S} = \iint \vec{F} \cdot \hat{n} \, dS.$$

In this drawing, the flux of $\vec{F}$ across the oriented surface element $d\vec{S}$ is represented by the volume of the gray boxes. You can see that the flux is largest when $\vec{F}$ and $d\vec{S}$ are parallel, but decreases as the angle between the vectors increases. In detail, the differential of flux is equal to the dot product between $\vec{F}$ and $d\vec{S}$ and the flux through an entire surface is $$\iint \vec{F} \cdot d\vec{S} = \iint \vec{F} \cdot \hat{n} \, dS.$$