Question

Find a parametric representation for the surface. the part of the plane z = x + 3 that lies inside the cylinder x2 + y2 = 9. (enter your answer as a comma-separated list of equations. let x, y, and z be in terms of s and/or θ.)

Answer 1

The cylinder is a clue to use cylindrical coordinates. Taking $x=3\cos\theta$ and $y=3\sin\theta$, we then are forced to use $z=3\cos\theta+3$. So the parameterization of the intersection of the plane and cylinder is

$\mathbf s(\theta)=(3\cos\theta,3\sin\theta,3\cos\theta+3)$

To get the surface, we can introduce a second parameter $r$ that "contracts" the elliptical intersection to a point. The simplest way to do this is to use

$\mathbf s(r,\theta)=(3r\cos\theta,3r\sin\theta,3r\cos\theta+3)$

with $0\le r\le1$ and $0\le\theta\le2\pi$.

Answer 2

The parameterized description would be a very common way of specifying both a surface as well as an implicit depiction. Coatings in two of the most important vector calculus theorems, the Stokes theorem as well as the divergence theorem often take a parameterized form, and its calculation can be defined as follows:

The part of the plane z = x +3 that lies inside cylinder  $x^2 + y^2 = 9$

Given that z is a function of x. z=x+3

The parameterization is $\vec{r}(x,y) = x\vec{i} + y\vec{j}+(x+ 3) \vec{k}$  with restriction $x^2 + y^2 \leq 9$

In polar form $\vec{r} (s, \theta)=s \cos \theta \vec{i}+s \sin \theta \vec{j} + (s \cos \theta +3)\vec{k}$

With $0 \leq \theta \leq 3\pi \ \ \ and\ \ \ 0 \leq s \leq 3$

So the parametric representation for the required surface is

$x= s \cos \theta\\\\y=s \sin \theta\\\\z=s \cos \theta +3 \\\\with \ \ 0\leq \theta \leq 3 \pi\ \ and\ \ 0 \leq s \leq 3$

So, the final answer is "$\bold{3 s \cos \theta\ ,\ 3s \sin \theta\ , \ 3s \cos \theta +3}$".

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