Question

In Exercises \(13-18,\) use a computer algebra system to graph the surface represented by the vector-valued function. $$ \begin{array}{l} \mathbf{r}(u, v)=2 u \cos v \mathbf{i}+2 u \sin v \mathbf{j}+u^{4} \mathbf{k} \\\ 0 \leq u \leq 1, \quad 0 \leq v \leq 2 \pi \end{array} $$

Short answer

Due to the nature of this problem, a short answer cannot be provided without a graphical display. However, the graph should represent a surface generated by mapping the parameter domain via \( \mathbf{r}(u, v)\)

Step-by-step solution

Identify the Vector-Valued Function

The given vector-valued function is \( \mathbf{r}(u, v)=2u \cos v \mathbf{i}+2u \sin v \mathbf{j}+u^{4} \mathbf{k} \).

Convert the Vector-Valued Function into Parametric Form

Rewrite the vector-valued function into three separate parametric functions which are representations for x, y, and z in terms of u and v: \( x(u,v) = 2u \cos v\), \( y(u,v) = 2u \sin v\), and \( z(u,v) = u^{4} \).

Use a Computer Algebra System to Graph the Surface

Using the three parametric functions from step 2, and the given parameters \( 0 \leq u \leq 1 \) and \( 0 \leq v \leq 2\pi \), input these into a computer algebra system to create a graph that represents the surface of the vector-valued function. Depending on the system used, this might involve using dedicated graphing commands which plot parametric surfaces.

Key concepts

Parametric Equations

When learning about vector-valued functions, one of the key concepts to understand is the idea of parametric equations. Parametric equations allow us to represent a curve or a surface in three-dimensional space by using one or more parameters.

Take for example our function from the exercise: \( \mathbf{r}(u, v)=2u \cos v \mathbf{i}+2u \sin v \mathbf{j}+u^{4} \mathbf{k} \). This function provides a set of parametric equations which define a surface. The parameters here are \( u \) and \( v \), and they each run over an interval; \( u \) runs from 0 to 1 and \( v \) runs from 0 to \(2\pi\).

For every pair of \( u \) and \( v \) within these intervals, we get a point \( (x(u,v), y(u,v), z(u,v)) \) in space where:

• \( x(u,v) = 2u \cos v \)
• \( y(u,v) = 2u \sin v \)
• \( z(u,v) = u^{4} \)

These points altogether form the desired surface. The beauty of parametric equations is that they can elegantly describe complex curves and surfaces that might be difficult or even impossible to characterize with just a single function \( y=f(x) \).

Computer Algebra System

To visualize advanced mathematical concepts, such as vector-valued functions and parametric surfaces, a Computer Algebra System (CAS) is incredibly useful. This type of software not only performs symbolic mathematics but also helps in graphing complex functions.

A CAS can manipulate and solve equations, even those involving variables, with ease. For instance, when working with parametric equations, a CAS can quickly solve for a set of points that satisfy the equations over the specified intervals. This is precisely what we need when we're asked to graph the surface of a vector-valued function like \( \mathbf{r}(u, v) \) from the exercise.

By inputting the equations for \( x \) and \( y \) in terms of \( u \) and \( v \) into the CAS along with the corresponding intervals for \( u \) and \( v \), the CAS can generate a 3D graph. This graph is a visual representation that significantly aids in the comprehension of the function’s geometric properties. Students are usually guided through the CAS interface to use the correct syntax and commands for graphing, which are essential skills in modern mathematics education.

3D Surface Graph

Graphing is an integral part of understanding complex mathematical concepts as it provides a visual representation that can be interpreted intuitively. In the context of vector-valued functions and parametric equations, a 3D surface graph serves to illustrate the shape of the surface defined by these equations.

A 3D surface graph translates the points calculated from the parametric equations into a form we can see and analyze. For our function \( \mathbf{r}(u, v) \), a 3D graph helps to visualize how the surface changes as the parameters \( u \) and \( v \) vary. You can imagine the surface as a sheet in a higher-dimensional space that contorts and stretches according to the values produced by the equations.

Software used to generate a 3D surface graph from parametric equations often provides tools for rotating, zooming, and even animating the graph. These interactive capabilities allow students to gain deeper insights into the behavior and properties of the surface. Students should ensure that they set up the correct domain and range for \( u \) and \( v \) to guarantee the surface is plotted correctly over the intended intervals.