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Question

Familiarize yourself with parametric representations of practically important surfaces by deriving a representation $z=f(x, y) \text { or } g(x, y, z)=0 \text {. }$, by finding the parameter curves (curves $u=$ const and $v=$ const) on the surface and a normal vector $\mathbf{N}=\mathbf{r}_u \times \mathbf{r}_v$ of the surface. (Show the details of your work.) Cone $\mathbf{r}(u, v)=[u \cos v, \quad u \sin v, \quad c u]$.

Solution

VerifiedAnswered 2 years ago

Answered 2 years ago

Step 1

1 of 9### Representation :

We know that the parametric representation of a surface $S$ in space is

$\mathbf{r}(u,v)=x(u,v)\mathbf{i}+y(u,v)\mathbf{j}+z(u,v)\mathbf{k}$

Hence, for the given problem

$x(u,v)=u\cos v,~y(u,v)=u\sin v,~z=cu$

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