Definition:Flux/Informal Definition

Definition
Let $\mathbf V$ be a vector field which acts on a region of space $R$.

Let $S$ be a surface embedded in $R$.

The concept of flux is most easily pictured in the context of the flow of fluids.

Let there be a fluid whose flow is defined by the vector $\mathbf v$.

Imagine a small loop of wire enclosing an area element $\delta S$ placed in the fluid, perpendicular to the direction of $\mathbf v$.

The flux $F$ of fluid through $\delta S$ is the rate of flow of fluid through the loop.

In this case this is the product of speed and area:
 * $F = \size {\mathbf v} \delta S$

Now suppose $\delta S$ is rotated so that the normal of $\delta S$ is now at some angle $\psi$ to the direction of $\mathbf v$.

Looking along the direction of $\mathbf v$, the area through which the fluid can flow is now $\delta S \cos \phi$


 * $F = \size {\mathbf v} \delta S \cos \phi$

Hence by definition of dot product:
 * $F = \mathbf v \cdot \delta \mathbf S$

where $\delta \mathbf S$ is now considered as a vector area.