Question

Use a computer algebra system to graph the surface represented by the vector- valued function. $$ \begin{array}{l} \mathbf{r}(u, v)=2 \sinh u \cos v \mathbf{i}+\sinh u \sin v \mathbf{j}+\cosh u \mathbf{k} \\ 0 \leq u \leq 2, \quad 0 \leq v \leq 2 \pi \end{array} $$

Short answer

Create a plot in a computer algebra system using the transformed function: \(x = 2 \sinh u \cos v\), \(y = \sinh u \sin v\), and \(z = \cosh u\). Make sure to specify the ranges of the parameters: \(0 \leq u \leq 2\) and \(0 \leq v \leq 2 \pi\).

Step-by-step solution

Understanding the Function

The vector-valued function \(\mathbf{r}(u, v)\) is given by \(2 \sinh u \cos v \mathbf{i}+\sinh u \sin v \mathbf{j}+\cosh u \mathbf{k}\). Here, \(u\) and \(v\) are parameters that vary from 0 to 2 and 0 to \(2 \pi\) respectively. This function uses hyperbolic sine and cosine, along with sine and cosine functions for the parameterization of the surface.

Transforming the Function for the Computer Algebra System

In the computer algebra system, this function will be transformed into the coordinate system. The \(x\), \(y\), and \(z\) coordinates of a point on the surface are given by the \(i\), \(j\), and \(k\) components of the function respectively. So, \(x = 2 \sinh u \cos v\), \(y = \sinh u \sin v\), and \(z = \cosh u\).

Plotting the Surface

Most computer algebra systems have commands to plot surfaces. Use the commands specific to your system, inputting the expressions for \(x\), \(y\), and \(z\) from the previous step, and the ranges for \(u\) and \(v\). Make sure to use a sufficient number of points in each direction to accurately depict the surface.

Interpreting the Graph

Once the graph is created, observe it to understand the behavior of the function. The appearance of the graph will depend on the specificities of the function and the range of the parameters.