Integral surface $(S) =(x=u+2v,y= u^2-v^2,z= uv+5)$.

Question

Calculate the area of ​​the surface S defined by $$(S) =S=\begin{cases} & \text{ } x=u+2v \\ & \text{ } y= u^2-v^2 \\ & \text{ } z= uv+5 \end{cases}$$ where $$A(S)=\frac{1}{2}\int_{(C)} xdy.ydx$$ and $$(u,v)\in R^2$$.I think of Green-Riemann Theorem but how to know if the curve is a closed one?

Answer 1

Hint:

Use the area formula of a parametric surface in the vector form $$\mathbf{r}=(x,y,z)=(u+2v,u^2-v^2,uv+5)$$ and $$A=\iint_{D}\left| \frac{\partial\mathbf{r}}{\partial u} \times \frac{\partial\mathbf{r}}{\partial v}\right|\,du\,dv.$$ but here we need $D$ to find $A$!