Problem 19

Question

Plot the parametric surface over the indicated domain. $\mathbf{r}(u, v)=2 \cos v \mathbf{i}+3 \sin v \mathbf{j}+u \mathbf{k} ;-6 \leq u \leq 6$, $0 \leq v \leq 2 \pi$

Step-by-Step Solution

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Answer

Plot elliptical circles on the z-axis from $-6$ to 6 for $v$ in $[0, 2\pi]$.

1Step 1: Understand Parametric Surface Functions

A parametric surface is defined by a vector function, $\mathbf{r}(u, v)=2 \cos v \mathbf{i}+3 \sin v \mathbf{j}+u \mathbf{k}$, where $u$ and $v$ are parameters that vary within given domains. The function is composed of three components: $x(u, v) = 2 \cos v$, $y(u, v) = 3 \sin v$, and $z(u, v) = u $. These components can describe points on a surface as $u$ and $v$ change.

2Step 2: Determine the Domain for Parameters

The given ranges for the parameters are $-6 \leq u \leq 6$ and $0 \leq v \leq 2\pi$. This means $u$ moves between negative six and six along the vertical $z$-axis, while $v$ revolves from zero to $2\pi$ radians to complete a full circle on the $xy$-plane.

3Step 3: Plot the Surface in Cartesian Coordinates

Convert the parametric equations into a surface by considering how $(x, y, z)$ coordinates change with $u$ and $v$. Plot these equations: $x = 2 \cos v$, $y = 3 \sin v$, and $z = u$. For each value of $u$ within its range, evaluate $\cos v$ and $\sin v$ over $v$'s domain, which forms consecutive elliptical circles along the $z$-axis.

4Step 4: Visualize the Surface

Using visual tools like graphing calculators or software (e.g., Python's Matplotlib or MATLAB), input the parametric equations. For every $v$ from $0$ to $2\pi$, these functions trace out ellipses of semi-major axis 3 ($y$)-axis) and semi-minor axis 2 ($x$-axis) at each height $u$. Continue this process for $u$ from $-6$ to 6 to plot the entire surface.

Key Concepts

Vector FunctionCartesian CoordinatesDomain of ParametersGraphing Software

Vector Function

A vector function, in the context of parametric surfaces, describes how points on a surface can be represented using parameters. In our example, the vector function is given by: \[ \mathbf{r}(u, v)=2 \cos v \mathbf{i}+3 \sin v \mathbf{j}+u \mathbf{k} \]This function has three components:

$x(u, v) = 2 \cos v$
$y(u, v) = 3 \sin v$
$z(u, v) = u$

These components show how each point on the surface changes as the parameters $u$ and $v$ vary. The vector function essentially bundles these components to create a path or surface in three-dimensional space. This particular function forms elliptical circles with their centers moving along the $z$-axis.

Cartesian Coordinates

Cartesian coordinates allow us to describe every point in a three-dimensional space with three numbers $(x, y, z)$. In parameterization, the coordinates are derived from vector functions, transforming parametric functions into concrete locations in space. In our example, the vector function breaks down into Cartesian coordinates as follows:

$x = 2 \cos v$ controls the horizontal stretch in the $x$-direction.
$y = 3 \sin v$ determines the stretch in the $y$-direction.
$z = u$ indicates height along the $z$-axis.

By evaluating these functions over their respective parameter domains, we can generate the surface in three-dimensional space. Visualizing surface changes in terms of these Cartesian coordinates aids in understanding the geometric nature of the shape described by the parametric function.

Domain of Parameters

The domain of parameters defines the range of values that parameters $u$ and $v$ can take and how they influence the parametric surface. In our given problem:

$-6 \leq u \leq 6$ indicates the range along the vertical $z$-axis, showing how far up and down the surface extends.
$0 \leq v \leq 2\pi$ represents a full circle's progression on the $xy$-plane, capturing the rotation around the center.

Within these domains, the parameters tailor the configuration and extent of the parametric surface. Understanding these constraints is essential as they guide how the surface can be plotted and where each point will land in our three-dimensional space.

Graphing Software

Using graphing software to plot parametric surfaces is a powerful way to visualize mathematical structures. Calculators and software like Python's Matplotlib or MATLAB allow you to input parametric equations and render the resulting surfaces. When graphing the example provided:

Begin by defining the range for parameter $u$, from $-6$ to $6$.
Set the range for parameter $v$, from $0$ to $2\pi$.
Calculate $x = 2 \cos v$ and $y = 3 \sin v$ for each point $(u, v)$.
Plot these points in a 3D graph as an elliptical path adjusted along the $z$-axis by $u$.

These steps create a visual representation of the function, aiding our understanding and allowing us to explore the surface's properties and features more intuitively.

Problem 19

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