Flux

The amount of a vector field that is passing through a surface

Formula

$$\Phi =\int {\int }_S\vec{F}\cdot \hat{n}dS$$

Requirements

1. $S$ is a smooth surface
2. $S$ is orientable (continuous $\hat{n}$)
   • Means it has 2 sides
   • $\pm$ of $\hat{n}$ is the orientation of the surface (which direction you want to measure flow in)

Properties

• ${\Phi }_{\uparrow }=-{\Phi }_{\downarrow }$ (2D)
• ${\Phi }_{\text{out}}=-{\Phi }_{\text{in}}$ (3D)

Solving

Across a Graph

• For
  • $z=g(x,y)$
  • $\vec{n}=\pm ⟨g_x^{\prime },g_y^{\prime },-1⟩$

$$\Phi =\int {\int }_D{\vec{F}\cdot \vec{n}|}_{z=g(x,y)}dA$$

Or use the Divergence Theorem

Parametric Surface

• For
  • $\vec{r}(t,s)$
  • $\vec{n}={\vec{r}}_t^{\prime }\times {\vec{r}}_s^{\prime }$

$$\Phi =\int {\int }_D\vec{F}(\vec{r}(t,s))\cdot ({\vec{r}}_t^{\prime }\times {\vec{r}}_s^{\prime })dA$$