How can we approximate a patch by $r_{u}$,$r_{v},$\Delta u$,$\Delta v$

Question

I am having the problem understanding the fact that the area of the patch can be approximated by$$|(\Delta u r_{u})\times (\Delta v r_{v})=|r_{u}\times r_{v}| \Delta u \Delta v$$ I don't understand how this came to be? won't the magnitude of $r_{u}$ or $r_{v}$ affect the approximation. I think it should be divided by its magnitudes? I am confused...

Answer 1

I am not sure this is what you want, but the area of a parallelogram ABCD is given by $$ ||\vec{AB}\times \vec{AD} || $$ Therefore, given a surface $S$ with parametrization $$ \vec{r}(u,v)=(x(u,v),y(u,v),z(u,v)),\quad (u,v)\in D $$ $ ||\vec{r}_u\times \vec{r}_v|| $ is an infinitisemal area of $S$ around the point $\vec{r}(u,v)$. To obtain the area of the whole surface, you need to sum these small areas over all points of $D$, that is $$ A(S)=\iint_D ||\vec{r}_u\times \vec{r}_v||\; dudv $$

Another way of looking at it is by considering the one dimensional case. Instead of dealing with a surface, consider a curve $C$ with parametrization $$ \vec{r}(t)=(x(t),y(t),z(t)),\quad t\in [a,b] $$ As one knows, the length of this curve equals $$ L( C)= \int_a^b ||\vec{r}'(t)||\; dt, $$ or, to use the same notations as before: $$ L( C)= \int_a^b ||\vec{r}_t||\; dt. $$ See how the structure of the formula is the same?