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

vousetmoiec 2021-11-12 Answered
Find a vector-valued function whose graph is the indicated surface.
The cylinder \(\displaystyle{4}{x}^{{2}}+{y}^{{2}}={16}\)

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Expert Answer

Helen Rodriguez
Answered 2021-11-13 Author has 8515 answers

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
Answer:
\(\displaystyle{r}{\left({u},{v}\right)}={2}{\cos{{u}}}{i}+{4}{\sin{{u}}}{j}+{v}{k}\)

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