Flux

A flux integral refers to a line integral or surface integral where the integrand is:

$$\vec{F}\cdot {\vec{n}}_s$$

where $\vec{F}$ is a vector field taken in the dot product with the unit normal vector.

For line integrals, we represent over an open contour and closed contour, respectively:

$${\int }_a^b\vec{F}\cdot {\vec{n}}_{s}dl$$ $${\oint }_C\vec{F}\cdot {\vec{n}}_{s}dl$$

For surface integrals over an open and closed surface:

$${\iint }_S\vec{F}\cdot {\vec{n}}_{s}dS$$ $$\oint {\oint }_S\vec{F}\cdot {\vec{n}}_{s}dS$$

Geometrically speaking, the flux quantifies how much of the vector field leaves or enters through the surface or contour.

Computations

We always start by finding a parameterisation, then writing $\vec{F}$ in terms of that. Helpfully to find $\hat{n}dS$ we can use:

$$\hat{n}dS=\frac{\partial \vec{r}}{\partial u}\times \frac{\partial \vec{r}}{\partial v}$$

Then we take the dot product of the parameterised representation of $\vec{F}$ with $\hat{n}dS$.

For a vector field $\vec{F}=⟨x,y⟩$, if we re-write as a complex-valued function $\bar{f}=x-iy$ and evaluate the same contour integral, then the imaginary part is the flux.

See also

• Magnetic flux