Flux Integrals | NST Mathematics

Surfaces

A surface $S$ in 3D is often parameterised by:

$\mathbf{r}(u, v) = (x(u,v), y(u,v), z(u,v))$

Orientation: A surface has two sides. The normal vector $\mathbf{n}$ points to one side.

For a small patch of surface:

The flux of vector field $\mathbf{F}$ through surface $S$ :

$\Phi = \int_S \mathbf{F} \cdot d\mathbf{S} = \int_S \mathbf{F} \cdot \mathbf{n}\,dS$

$Surface S with normal vectors n, showing vector field F passing through and the flux integral interpretation$

Interpretation: $\mathbf{F} \cdot \mathbf{n}$ is the component of $\mathbf{F}$ through the surface. Added up over $S$ , this gives total flux.

Fluid flow: Flux = volume of fluid crossing surface per unit time
Electric field: Flux = $\int \mathbf{E} \cdot d\mathbf{S}$ (related to enclosed charge)
Heat flux: Rate of heat flow through surface

For $\mathbf{r}(u, v)$ :

$d\mathbf{S} = \frac{\partial \mathbf{r}}{\partial u} \times \frac{\partial \mathbf{r}}{\partial v}\,du\,dv$

Direction given by right-hand rule for $(u, v)$ coordinates.

For surface $g(x, y, z) = c$ :

$\mathbf{n} = \pm \frac{\nabla g}{|\nabla g|}$

Choose sign for correct orientation.

For $z = f(x, y)$ :

$\mathbf{n}\,dS = \left(-\frac{\partial f}{\partial x}, -\frac{\partial f}{\partial y}, 1\right)dx\,dy$

Find flux of $\mathbf{F} = (0, 0, z)$ through the hemisphere $x^2 + y^2 + z^2 = a^2$ , $z \geq 0$ , with outward normal.

On the hemisphere: The unit normal is $\hat{\mathbf{n}} = \frac{1}{a}(x, y, z)$ .

$\mathbf{F} \cdot \hat{\mathbf{n}} = \frac{z^2}{a}$

Flux = $\int_S \frac{z^2}{a}\,dS$

By symmetry, $\langle x^2 \rangle = \langle y^2 \rangle = \langle z^2 \rangle$ on the sphere, so $z^2 = \frac{a^2}{3}$ .

Surface area = $2\pi a^2$ .

Flux = $\frac{a^2/3}{a} \cdot 2\pi a^2 = \frac{2\pi a^3}{3}$ .

Orientation: Reversing normal multiplies flux by $-1$ .

Closed surfaces: For closed surface $S$ , by convention the normal points outward.

The vector area of a surface:

$\mathbf{S} = \int_S d\mathbf{S}$

Closed surface: Vector area = $\mathbf{0}$

(Each patch cancels with opposite contribution.)

Flux of $\mathbf{F}$ through flat surface $S$ in plane with normal $\mathbf{n}$ :

$\Phi = \mathbf{n} \cdot \int_S \mathbf{F}\,dS$

If $\mathbf{F}$ is constant: $\Phi = \mathbf{F} \cdot \mathbf{n} \cdot \text{Area}$