# Find a vector-valued function whose graph is the indicated surface. N

Find a vector-valued function whose graph is the indicated surface.
The cylinder $$\displaystyle{4}{x}^{{2}}+{y}^{{2}}={16}$$

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Step 1
In polar coordinates, $$x=r\cos\theta$$, $$y=r\sin\theta$$
Using this rule, using appropriate substitutions:
$$\displaystyle{x}={2}{\cos{{u}}}$$, $$\displaystyle{y}={4}{\sin{{u}}}$$
Step 2
Considering $$\displaystyle{z}={v}$$, the vector form is given by:
$$\displaystyle{r}{\left({u},{v}\right)}={2}{\cos{{u}}}{i}+{4}{\sin{{u}}}{j}+{v}{k}$$
Step 3
$$\displaystyle{r}{\left({u},{v}\right)}={2}{\cos{{u}}}{i}+{4}{\sin{{u}}}{j}+{v}{k}$$