Problem 107 - Thomas Calculus

Question

Symmetry What symmetry will you find in a surface that has an equation of the form \(r=f(z)\) in cylindrical coordinates? Give reasons for your answer.

Step-by-Step Solution

Answer

The surface is symmetric with respect to the z-axis.

Step 1: Understand the Equation

The equation provided is in the form of \(r=f(z)\). This means that the radial distance \(r\) is expressed as a function of \(z\), which is the height in cylindrical coordinates. Cylindrical coordinates have three variables: \(r\) (radial), \(\theta\) (angular), and \(z\) (height).

Step 2: Identify Potential Symmetries

In cylindrical coordinates, we typically analyze symmetry with respect to the \(z\)-axis, the origin in the r-\(\theta\) plane, and various planes like \(z=0\) or \(\theta=0\). The form \(r=f(z)\) suggests no dependency on \(\theta\).

Step 3: Check Symmetry with Respect to the z-Axis

Since the function \(r=f(z)\) does not involve \(\theta\), the surface must be symmetric about the \(z\)-axis. Any rotation around the \(z\)-axis (variations in \(\theta\)) will not change the form of the surface since \(r\) remains purely a function of \(z\).

Step 4: Evaluate the Equation in Different Planes

Since \(f(z)\) is independent of \(\theta\), it suggests symmetry in any plane \(\theta=\text{constant}\), meaning if you cut the surface with a plane along any vaule of \(\theta\), you will see a profile that is identical, centered on the \(z\) axis.

Key Concepts

Symmetry in Cylindrical Coordinates

When dealing with cylindrical coordinates, symmetry often plays a pivotal role in understanding the geometry of a surface. Cylindrical coordinates utilize three variables: radial distance (\(r\)), angular component (\(\theta\)), and vertical height (\(z\)). In the context of the equation \(r = f(z)\), which defines surfaces in cylindrical coordinates, recognizing symmetry becomes straightforward.

The equation \(r = f(z)\) indicates that the radial distance \(r\) is a function solely of \(z\), independent of \(\theta\). This suggests a strong symmetry about the \(z\)-axis because the value of \(r\) remains unchanged for any \(\theta\). You can imagine twisting or rotating the surface around the \(z\)-axis: nothing about the distance from the center changes.

• This axisymmetric property simplifies many problems in physics and engineering.
• It reduces complex three-dimensional investigations to two dimensions with respective cross-sectional views.

Utilizing symmetry simplifies analyzing and visualizing the surface, which can help in mathematical computations and structural understanding.

Understanding Radial Distance

Radial distance, commonly denoted as \(r\), is a crucial component in cylindrical coordinate systems. It measures the distance from a point to the \(z\)-axis within a plane parallel to the standard Cartesian plane.

Radial distance works similarly to how radius functions in a circle. In cylindrical coordinates, every point's position is defined by how far it stretches out radially from the central \(z\)-axis.

• In equations like \(r = f(z)\), the radial distance varies based on vertical height (\(z\)).
• Any changes in \(\theta\) (rotation) don't affect it.

This understanding highlights why \(r\) surfaces in such equations often form rotationally symmetric shapes, akin to tubes or cylinders growing out from the \(z\)-axis.

The Role of the z-Axis

The \(z\)-axis is the vertical line around which the other coordinates in a cylindrical system are measured. It acts as the axis of symmetry for surfaces described by equations like \(r = f(z)\).

Being independent of \(\theta\) ensures that the surface has uniformity when revolved around this axis, which makes it especially beneficial in problems involving revolution. This symmetry around the \(z\)-axis implies:

• The surface generated is rotationally invariant. This essentially means any section slice along \(\theta\) will appear identical.
• Engineering applications often use this principle by treating structures like pipes and beams under assumptions of uniform cross-sections.

The \(z\)-axis thus becomes a fundamental reference line, simplifying the conversion between cylindrical and Cartesian coordinates and aiding in the comprehension of various physical systems.