Problem 3 - 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.) Cone frustum The first-octant portion of the cone \(z=\) \(\sqrt{x^{2}+y^{2}} / 2\) between the planes \(z=0\) and \(z=3\)

Step-by-Step Solution

Answer

The parametrization is \((x, y, z) = (2z \cos \theta, 2z \sin \theta, z)\) for \(0 \leq z \leq 3\) and \(0 \leq \theta \leq \frac{\pi}{2}\).

Step 1: Identifying the Geometry

The frustum of a cone is given by the equation \(z = \frac{1}{2}\sqrt{x^2 + y^2}\) and is bounded by \(z = 0\) and \(z = 3\). Since it's in the first octant, \(x, y,\) and \(z\) must all be positive.

Step 2: Expressing in Cylindrical Coordinates

Since the problem involves a cone which is typically symmetric around the z-axis, we use cylindrical coordinates. We know \(x = r \cos \theta\), \(y = r \sin \theta\), and \(z = \frac{1}{2} \sqrt{x^2 + y^2} = \frac{r}{2}\). This means \(r = 2z\), and \(0 \leq \theta \leq \frac{\pi}{2}\) due to the first octant.

Step 3: Parametrizing the Surface

To parametrize the surface, let \(z\) be the parameter such that \(0 \leq z \leq 3\). The radius \(r\) is then \(2z\), and the parametrization is \((x, y, z) = (2z \cos \theta, 2z \sin \theta, z)\). \(\theta\) varies from \(0\) to \(\frac{\pi}{2}\) and \(z\) from \(0\) to \(3\).

Step 4: Conclusion and Confirming the Parametrization

The parametrization \(\textbf{r}(z, \theta) = (2z \cos \theta, 2z \sin \theta, z)\) correctly describes the first-octant portion of the cone frustum as it ensures positivity of \(x, y,\) and \(z\), and correctly adheres to the geometric shape from \(z = 0\) to \(z = 3\).

Key Concepts

Cylindrical Coordinates

Cylindrical coordinates provide a useful system for describing points in three-dimensional space where symmetry around the z-axis exists. In such a system, a point is represented by three values:

• Radial distance ( ): The distance from the point to the z-axis.
• Angular displacement (θ): The angle measured from the positive x-axis, similar to polar coordinates.
• Height (z): The elevation of the point above the xy-plane.

When using cylindrical coordinates to convert Cartesian coordinates \(x, y, z\) into cylindrical form, we utilize the following conversions:

• \(x = r \cos \theta\)
• \(y = r \sin \theta\)
• \(z = z\)

This is particularly advantageous when dealing with problems that possess radial symmetry, such as the cone frustum in the problem statement. By leveraging cylindrical coordinates, it becomes easier to express and manipulate the equations and constraints involving the surface geometries.

Cone Frustum

A cone frustum is the portion of a cone that lies between two parallel planes slicing through it. In the given problem, the frustum is defined by the equation \(z = \frac{1}{2} \sqrt{x^2 + y^2}\), bounded between \(z = 0\) and \(z = 3\). To visually imagine a cone frustum, picture a cone with its tip removed and the remaining parts forming a ring-like shape. In terms of parametrization, the values for \(z\) delineate its extent, while the radial coordinates describe its circular cross-section.
The calculations in cylindrical coordinates reveal that.

• The radius \(r\) of each circular cross-section shifts linearly with \(z\), given by \(r = 2z\) in this context.
• The height, \(z\), varies between the specified planes, i.e., \(0\) to \(3\).

This clear parametric relationship ensures that the surface of the cone frustum is represented accurately, with its circular symmetry maintained throughout.

First Octant

The concept of the first octant in a three-dimensional space pertains to the unique division where all the coordinate axes \(x, y,\) and \(z\) are positive. Imagine the 3D coordinate system divided into eight spaces like slices of a pie. The first octant is the space where imaginary lines from positive x, y, and z axes meet.
In many problems of three-dimensional geometry, constraints ensure that calculations or geometric representations are confined to this segment of space.

• This helps in simplifying representations and ensuring positive coordinate values, which can often align more naturally with real-world physical scenarios.
• In our context dealing with our cone frustum, these constraints make sure that \(r\) angle \(\theta\) are non-negative and feasible with the defined structure.
• By considering only the first octant, we assign \(\theta\) to vary between \(0\) and \(\frac{\pi}{2}\), representing only the first quadrant in polar terms.

Thus, working within the first octant keeps our geometric problem grounded in a practical and simplified context.