I'm doing some surface integrals and I'm confused as to which formula I should use.
Let's say you have a parametric representation of a surface S.
$\overrightarrow{r}(u,v) = x(u,v)\overrightarrow{i} + y(u,v)\overrightarrow{j} + z(u,v)\overrightarrow{k}$
That is, from what I can gather, taking some 2-dimensional representation of a surface and extending it outward into a 3-dimensional space. And the equation above gives us the vector $\overrightarrow{r}$ that gives the x, y, and z coordinates of the point on the 3D surface.
After computing two tangent vectors, $r_u$ and $r_v$, you can take the cross product to get the area of a parallelogram spanned out by those two vectors.
$\iint_D ||r_u \times r_v||~dA$
This is supposed to give us the surface area of that 2D surface that has been extended out into the third dimension. Right?
But where I get confused is when my text introduces a completely different equation for the surface area for a surface z = g(x,y).
$\iint_S f(x,y,z)dS = \iint_D f(x,y,g(x,y))\sqrt{(\frac{\partial g}{\partial x})^2 + (\frac{\partial g}{\partial y})^2 + 1} dA$
When are you supposed to use each of these equations? Is the first when you have a two dimensional surface that is extended out into the third dimension, and the second one is when you have just a straight up 3D surface?