I am studying for a Calculus exam, and one of the topics I should know about is surface integrals. Now, I am using Stewart 6e, and in there I have found several equations for computing surface integrals. These are the equations I found:
$\iint_{S}{f(x, y, z)\:dS}=\iint_{D}{f(\vec{r}(u,v))*\left|\vec{r}_{u}\times\vec{r}_{v}\right|\:dA} \\ \iint_{S}{f(x, y, z)\:dS}=\iint_{D}{f(x, y, g(x, y))*\sqrt{\left(\frac{dg}{dx}\right)^{2}+\left(\frac{dg}{dy}\right)^{2}+1}\:dA} \\ \iint_{S}{\vec{F}\:d\vec{S}}=\iint_{D}{\vec{F}*\left(\vec{r}_{u}\times\vec{r}_{v}\right)\:dA} \\ \iint_{S}{\vec{F}\:d\vec{S}}=\iint_{D}{\left(-P\left(\frac{dg}{dx}\right)-Q\left(\frac{dg}{dy}\right)+R\right)\:dA}$
So, my question is: How do I know which one to use for a given exercise? Obviously the bottom two should be used if I'm dealing with a vector field, and the top two if I'm dealing with just an equation. Also, I noticed that the second and last one both require a seperate function g(x,y). But still, it isn't quite clear to me when to use which one. Please help me out!