Question

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

Short answer

The vector-valued function for the given cylinder \(x^{2}+y^{2}=16\) is \(\pmb{r}(\theta, z) = 4cos(\theta) \pmb{i} + 4sin(\theta) \pmb{j} + z \pmb{k}\)

Step-by-step solution

- Convert to cylindrical coordinates

In order to represent the cylinder as a vector-valued function, we will need to convert the given equation from Cartesian to cylindrical coordinates. The standard representation in cylindrical coordinates consists of radian \(r\), angle \(\theta\), and the height \(z\). Here, the radius of the cylinder is represented as \(r = 4\), as by taking the square root of 16 we get 4. Therefore \(x = rcos(\theta)\) and \(y = rsin(\theta)\)

- Create vector-valued function

Given the Cartesian coordinates figurative presentation and their relationship to cylindrical coordinates as shown in Step 1, we can create the vector-valued function that represents the given cylinder. This type of function often takes two parameters - commonly used are \(r\), \(\theta\), and \(z\). For this cylinder, However, we know that the radius r is fixed as 4. Therefore, the two variables required are \(\theta\) and \(z\). Thus the vector-valued function is \(\pmb{r}(\theta, z) = 4cos(\theta) \pmb{i} + 4sin(\theta) \pmb{j} + z \pmb{k}\)

- Verify the vector-valued function

The given vector-valued function should produce the cylinder when graphed. The function contains the cos and sin functions for the i and j components, representing the circular cross-section of the cylinder, multiplied by 4 which represents the radius of the cylinder. The k component represents the height of the cylinder which could be any real number. This makes sense as a cylinder extends infinitely in the z-direction