# Definition:Flux

## 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 flux through $S$ is the dot product of $\mathbf V$ with the vector area of $S$.

Hence flux is a scalar quantity.

### Informal Definition

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.

### Direction

While flux is a scalar quantity, it does have a direction.

The sign of vector area $\delta S$ through which the flux operates depends upon the right-hand rule according to the direction of rotation around $S$ when describing it.

Hence, having decided on the sign of $\delta S$, the flux through $\delta S$ is either:

positive, where the angle $\psi$ between the normal to $\delta S$ is acute
negative, where the angle $\psi$ between the normal to $\delta S$ is obtuse.

That is:

a positive flux when $\delta S$ is oriented from left to right

is:

a negative flux when $\delta S$ is oriented from right to left.

### Total Flux

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

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

The total flux through $S$ is the surface integral over $S$ of the dot product of $\mathbf V$ with the vector area of $S$:

$F = \ds \int_S \mathbf V \cdot \rd S$

where $\d S$ is an infinitesimal area element of $S$.

## Illustration $\qquad$ As can be seen, more flux passes through $\delta S$ when it is positioned perpendicular to the direction of $\mathbf v$, as in the diagram on the left, than when not, as in the diagram on the right.

## Also see

• Results about flux can be found here.