Surface Integrals

Multivariable Surfaces

$$\begin{array}{rl}& S:z=g(x,y),(x,y)\in D\\ & {\iint }_{S}f(x,y,z)ds={\iint }_{D}f(x,y,g(x,y))\sqrt{1+(g_x^{\prime }{)}^2+(g_y^{\prime }{)}^2}dA\end{array}$$

Parametric Surfaces

$$\begin{array}{rl}& S:\vec{r}(u,v),(u,v)\in D\\ & {\iint }_{S}f(\vec{r})ds={\iint }_{D}f(\vec{r}(u,v))\Vert {\vec{r}}_u^{\prime }\times {\vec{r}}_v^{\prime }\Vert dA\end{array}$$