Problem 11 - Thomas Calculus

Question

Find a parametrization of the surface. (There are many correct ways to do these, so your answers may not be the same as those in the back of the book.) Circular cylinder band The portion of the cylinder \(y^{2}+z^{2}=9\) between the planes \(x=0\) and \(x=3\)

Step-by-Step Solution

Answer

The parametrization is \(\mathbf{r}(x, \theta) = (x, 3\cos(\theta), 3\sin(\theta))\), where \(x\in [0, 3]\) and \(\theta\in [0, 2\pi]\).

Step 1: Identify the shape of the surface

The given equation \(y^2 + z^2 = 9\) represents a circular cylinder with radius 3, centered on the x-axis. We need to parametrize this surface between the planes \(x=0\) and \(x=3\). The cylinder's cross-section in the yz-plane (circle of radius 3) suggests using trigonometric functions for parametrization.

Step 2: Choose parameters for the surface

To parametrize the circular cross-section, we'll use an angle \(\theta\) to describe points on the circle in the yz-plane. The other variable will be \(x\), which ranges from 0 to 3. Thus, we'll use \(\theta\) and \(x\) as our parameters.

Step 3: Write the parametrization using the chosen parameters

We express \(y\) and \(z\) using \(\theta\), as they form a circle: \(y = 3\cos(\theta)\) and \(z = 3\sin(\theta)\). Therefore, a parametrization of the surface is: \(\mathbf{r}(x, \theta) = (x, 3\cos(\theta), 3\sin(\theta))\), where \(x\in[0, 3]\) and \(\theta\in[0, 2\pi]\).

Step 4: Verify the parametrization fits the given constraints

Check that the parametrization covers the entire surface of the cylinder band. For all \(x\) between 0 and 3, and \(\theta\) between 0 and \(2\pi\), the circle \(y^2 + z^2 = 9\) holds true, and \(x\) covers the range from \(x=0\) to \(x=3\). Hence, the parametrization is valid for describing the surface between the specified planes.

Key Concepts

Understanding the Circular Cylinder

A circular cylinder is a three-dimensional geometric shape with two parallel circular bases and a curved surface connecting them. In the given exercise, the equation \( y^2 + z^2 = 9 \) describes a circular cross-section. This equation indicates that we have circles lying parallel to the x-axis with radius 3. Each cross-section of the cylinder, therefore, maintains a constant radius and forms perfect circles when visualized in the yz-plane.

When dealing with the circular cylinder, it's essential to remember:

• The axis of the cylinder is the straight line connecting the centers of circular cross-sections.
• In this problem, the cylinder is aligned along the x-axis.
• The planes \( x=0 \) and \( x=3 \) confine our focus to only part of this cylinder, creating a band.

Understanding this setup is important before attempting to define any parametrization of the cylinder.

Trigonometric Parametrization

Trigonometric parametrization is a technique used to express circular shapes using trigonometric functions. For our circular cylinder band, the circle in the yz-plane is defined using the angle \( \theta \), where the component formulas are \( y = 3\cos(\theta) \) and \( z = 3\sin(\theta) \).

Here's why this works:

• The trigonometric identities \( \cos^2(\theta) + \sin^2(\theta) = 1 \) give us the circular property needed (the basis of how circles are constructed in trigonometric terms).
• Multiplying these identities by 3 ensures they match the given radius of the circle.
• This parametrization implies that as \( \theta \) moves from 0 to \( 2\pi \), it draws a complete circle.

Combining these expressions with \( x \), which is free to vary between 0 and 3, gives a full trigonometric parametrization of the section of the cylinder.

Multivariable Calculus and Surface Parametrization

Surface parametrization is a powerful method in multivariable calculus that allows us to describe surfaces in space using two parameters. In this exercise, \( x \) and \( \theta \) are our parameters, creating a mapping of values in the domain onto the cylinder's curved surface.

Key points of surface parametrization include:

• The choice of parameters can often make calculations simpler and provide insight into the geometry of the surface.
• Parametrizations must respect the geometry they describe—in this case, ensuring all points satisfy \( y^2 + z^2 = 9 \).
• This method is crucial for computing surface areas, volumes, and performing visualizations in multivariable calculus.

Thus, understanding and creating parametrizations like the one defined here lays foundational knowledge in analyzing and interpreting more complex surfaces and shapes in three-dimensional space.