Surface Integrals

$$\iint_{S}f(x,y,z) ,\text{dS}$$

Parameterization

$$f(x,y,z)$$ $$\vec{r} = [x(u,v),\ y(u,v),\ z(u,v)]$$ $$ \text{dS} = |\vec{r_{u}} \times \vec{r_{v}}|\ \text{du}\ \text{dv}, \ \text{where} \begin{cases} \vec{r_{u}} = \left[\frac{\partial x}{\partial u},\ \frac{\partial y}{\partial u},\ \frac{\partial z}{\partial u}\right] \ \vec{r_{v}} = \left[\frac{\partial x}{\partial v},\ \frac{\partial y}{\partial v},\ \frac{\partial z}{\partial v}\right] \ \end{cases} $$ $$ \Rightarrow \iint_{S} f(\vec{r}) \cdot |\vec{r_{u}} \times \vec{r_{v}}|\ \text{du}\ \text{dv} $$

Surface Area of Parameterization

$$ \iint_{S} |\vec{r_{u}} \times \vec{r_{v}}|\ \text{du}\ \text{dv} $$

With dependent z

For cases where $$z = g(x, y) \Rightarrow \vec{r} = [x, y ,z(x,y)]$$ Here we can use this formula: $$\iint f[x,y,z(x,y)] \bullet \sqrt{\left(\frac{\partial z}{\partial x}\right)^{2}, \left(\frac{\partial z}{\partial y}\right)^{2} + 1}\ \text{dx}\ \text{dy}$$

Flux

The amount of the vector field $F$ that passes through the surface $S$. $$\int\int_{S} \vec{F} \times \vec{n}\ \text{dS}$$ $$\vec{n} = \frac{\vec{r_{u}} \times \vec{r_{v}}}{|\vec{r_{u}} \times \vec{r_{v}}|}$$ $$\Rightarrow \iint_{S} \vec{F} \cdot (\vec{r_{u}} \times \vec{r_{v}})\ \text{du dv}$$