Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

Plot the parametric surface over the indicated domain. \(\mathbf{r}(u, v)=u \mathbf{i}+3 \sin v \mathbf{j}+5 \cos v \mathbf{k} ;-6 \leq u \leq 6\), \(0 \leq v \leq 2 \pi\)

Step-by-Step Solution

Answer

Plot an elliptical cylinder extending along the x-axis using the given ranges for u and v.

Step 1: Understand the Parametric Equations

Recognize that the parametric surface is given by \[ \mathbf{r}(u, v) = u \mathbf{i} + 3 \sin v \mathbf{j} + 5 \cos v \mathbf{k} \]where the parameter \(u\) varies from \(-6\) to \(6\), and \(v\) varies from \(0\) to \(2\pi\). This represents the coordinates of points on a surface in 3D space, where \(x = u\), \(y = 3 \sin v\), and \(z = 5 \cos v\).

Step 2: Determine the Shape

Recognize that \(x = u\) is a linear function of \(u\), suggesting the surface extends along the x-axis. The equations \(y = 3 \sin v\) and \(z = 5 \cos v\) describe an ellipse in the yz-plane centered at the origin with semi-major axis 5 and semi-minor axis 3, as \(v\) sweeps from \(0\) to \(2\pi\).

Step 3: Visualize the Surface

Combine the understanding from Step 2: for each \(u\), the yz-component traces an elliptical motion while \(u\) acts as an independent motion along the x-axis. This results in a cylinder with elliptical cross-sections parallel to the yz-plane.

Step 4: Use a Plotting Software/Tool

Utilize software like MATLAB, Python (matplotlib), or any graphing tool capable of 3D plotting. Input the parametric equations:\[ x = u, \quad y = 3 \sin v, \quad z = 5 \cos v \]within the given domains: \(-6 \leq u \leq 6\) and \(0 \leq v \leq 2\pi\).

Step 5: Plot the Surface

Generate the plot by varying \(u\) and \(v\) within their domains. The resulting graph should show the characterized elliptical cylinder extending along the x-axis, outlined over the interval from \(-6\) to \(6\).

Key Concepts

3D Plotting

Understanding 3D plotting is essential when working with parametric surfaces. In 3D plotting, we create visual representations of objects within a three-dimensional space using a set of three coordinates: x, y, and z. This can help students grasp the spatial relationships between different mathematical equations and their corresponding surfaces.

When plotting a parametric surface, it's crucial to account for both parameters and how they define different points on the plotted shape. By varying each parameter, we observe how the shape evolves over a continuous region within 3D space. For the given exercise, the parametric surface is defined by the vectors \( \mathbf{r}(u, v) = u \mathbf{i} + 3 \sin v \mathbf{j} + 5 \cos v \mathbf{k}\), representing points that form a complete surface.

To successfully create a 3D plot:

• Use graphing software like MATLAB or Python's matplotlib.
• Input the parametric equations into the software.
• Define the domain of parameters \(u\) and \(v\).
• Observe the rendered 3D surface, helping to visualize its features and characteristics.

These steps will guide students in better visualizing mathematical objects within three-dimensional settings, turning complex ideas into more tangible insights.

Elliptical Cylinder

An elliptical cylinder is a 3D shape that serves as a central concept in understanding the given exercise. Unlike a perfect circular cylinder, which has circular cross-sections, an elliptical cylinder showcases elliptical cross-sections, adding a twist to its geometry.

In the context of the task:

• The variable \(x = u\) indicates that the cylinder extends horizontally along the x-axis.
• The equations \(y = 3 \sin v\) and \(z = 5 \cos v\) describe an ellipse in the yz-plane.
• The semi-major axis length is 5, and the semi-minor axis length is 3.

Understanding these components will help students visualize how the elliptical shape repeats as the "slice" moves along the x-axis, forming a cylinder. By translating the concept of an ellipse into a three-dimensional framework, it becomes apparent that the elliptical nature persists along the primary linear axis defined by \(x = u\). This geometry commonly appears in advanced applications, including architecture and engineering, where non-circular cylinders provide applied solutions.

Parametric Equations

Parametric equations are a powerful tool in mathematics, allowing the representation of complex curves and surfaces in 3D space. Unlike standard equations which rely strictly on x, y, and z coordinates, parametric equations use parameters \(u\) and \(v\) to define these coordinates as a function of those parameters.

For this exercise, the parametric equations given are:

• \(x = u\)
• \(y = 3 \sin v\)
• \(z = 5 \cos v\)

These equations allow the formation of a surface by varying \(u\) and \(v\) within specified ranges. This approach is especially beneficial when working with surfaces that aren't easily described with conventional explicit functions.

Through parametric equations, students can uncover:

• How different functions contribute to the shape of a surface.
• The relationships between variables and parameters.
• An innovative way to explore and model 3D geometrical forms.

Using parametric equations simplifies complex geometries, offering insights into the logic of how surfaces behave in space, well beyond what traditional modeling would easily allow.