## Related questions with answers

Familiarize yourself with parametric representations of important surfaces by deriving a representation, by finding the parameter curves (curves u=const and v=const) of the surface and a normal vector N=ru*rv of the surface. Show the details of your work. xy-plane r(u, v)=(u, v)(thus ui+vj)

Solution

VerifiedThe paramater curves of the surface in the $xy$-plane parametrised by

$\pmb{r}(u,v) = u \cdot \pmb{i} + v\cdot \pmb{j} = (u,v)$

are:

$u=u_0( = const): \,\, r(u_0,v) = u_0 \cdot \pmb{i} + v \cdot \pmb{v} = (u_0,v)$

and

$v=v_0( = const): \,\, r(u ,v_0) = u \cdot \pmb{i} + v_0 \cdot \pmb{v} = (u,v_0)$

Notice that the first family of parameter curves is the family of all lines of the form $x=u_0$, while the second family of parameter curves is the family of all lines of the form $y=v_0$.

We have

$\pmb{r}_u = (1,0) = \pmb{i}, \,\, \pmb{r}_v = (0,1) = \pmb{j}$

It is easy to see that the normal vector is given by

$\pmb{N} = \pmb{r}_u \times \pmb{r}_v = \pmb{i} \times \pmb{j} = \pmb{k}$

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