Surface Integral

Scalar Surface Integral

A scalar surface integral computes a overall value across a surface, based on instantaneous concentrations at each $f(x,y,z)$, e.g. overall mass from instantaneous densities

$${\iint }_{S}f(x,y,z)dS$$

Explicit Surface

When $S$ can be expressed as $z=g(x,y)$

$${\iint }_{S}f(x,y,z)dS={\iint }_{D}f(x,y,g)\sqrt{g_x^2+g_y^2+1}dA$$

Parametric Surface

When $S$ can be expressed as $\vec{r}=<x(u,v),y(u,v),z(u,v)>$

$${\iint }_{S}f(x,y,z)dS={\iint }_{D}f(\vec{r}(u,v))|{\vec{r}}_u\times {\vec{r}}_v|dA$$

Vector Surface Integral

A vector surface integral gives you the flux over the surface. Flux is a measure of how much the vectors throughout the vector field are passing through the surface, along the normal direction.

$${\iint }_S\vec{F}\cdot d\vec{S}={\iint }_S\vec{F}\cdot \vec{n}dS$$

where the normals are outwardly oriented

Explicit Surface

When $S$ can be expressed as $z=g(x,y)$, and $\vec{F}=<P,Q,R>$

$${\iint }_S\vec{F}\cdot d\vec{S}={\iint }_D-P{g}_x-Q{g}_y+RdA$$

Parametric Surface

When $S$ can be expressed as $\vec{r}=<x(u,v),y(u,v),z(u,v)>$

$${\iint }_S\vec{F}\cdot d\vec{S}={\iint }_D\vec{F}\cdot ({\vec{r}}_u\times {\vec{r}}_v)dA$$