Flux | Sahithyan's Notes

The flux of a vector field through an oriented surface measures the net amount of flow passing through the surface in the chosen normal direction, assuming a fluid in steady state and incompressible. The volume of fluid flowing through a portion of a surface depends on:

Speed of the fluid.
Orientation of the surface relative to the flow.
Area of the portion of the surface.

Let $\boldsymbol{F}$ be a continuous vector field on an oriented surface $\sigma$ , and let $\hat{n}$ be the chosen unit normal field on $\sigma$ . The flux of $\boldsymbol{F}$ across $\sigma$ is

\Phi = \iint_\sigma \boldsymbol{F} \cdot \hat{n}\,\text{d}S

Flux through a Parametric Surface

Let $\sigma$ be a smooth parametric surface $\boldsymbol{r} = \boldsymbol{r}(u,v)$ , where $(u,v)$ varies over a region $R$ in the $uv$ -plane. Suppose the component functions of $\boldsymbol{F}$ are continuous on $\sigma$ and that

\frac{\partial \boldsymbol{r}}{\partial u} \times \frac{\partial \boldsymbol{r}}{\partial v}

determines the chosen orientation of $\sigma$ . Then

\Phi = \iint_\sigma \boldsymbol{F} \cdot \hat{n}\,\text{d}S = \iint_R \boldsymbol{F}(\boldsymbol{r}(u,v)) \cdot \left(\frac{\partial \boldsymbol{r}}{\partial u} \times \frac{\partial \boldsymbol{r}}{\partial v}\right) \text{d}A_{uv}

Flux through a Graph

Let $\sigma$ be a smooth surface of the form $z = g(x,y)$ , $y = g(x,z)$ , or $x = g(y,z)$ , written as $G(x,y,z) = 0$ by moving all terms to the left side, with orientation determined by $\nabla G$ . Let $R$ be the projection of $\sigma$ onto the coordinate plane determined by the independent variables of $g$ . If $\boldsymbol{F}$ is continuous on $\sigma$ ,

\Phi = \iint_\sigma \boldsymbol{F} \cdot \hat{n}\,\text{d}S = \iint_R \boldsymbol{F} \cdot \nabla G\,\text{d}A