Problem 89 - 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 has rotational symmetry around the \( z \)-axis.

Step 1: Understanding the Equation

The equation provided is in the form of cylindrical coordinates, where \( r = f(z) \). Here, \( r \) is a function of \( z \), indicating how the radius varies with the height \( z \). This suggests that the radial distance depends solely on the vertical coordinate \( z \).

Step 2: Analyzing Symmetrical Features

In cylindrical coordinates, if \( r \) depends only on \( z \), it implies that the surface is symmetrical around the \( z \)-axis. This is because, for any fixed \( z \), \( r \) will be the same for all values of the angular coordinate \( \theta \). Thus, the surface will look the same when rotated around the \( z \)-axis.

Step 3: Conclusion on Type of Symmetry

Based on the analysis, the surface described by the equation \( r = f(z) \) possesses rotational symmetry about the \( z \)-axis. This means that for every height \( z \), the cross-section is a circle centered on the \( z \)-axis.

Key Concepts

Cylindrical Coordinates

Cylindrical coordinates are an extension of polar coordinates into three dimensions. They are particularly useful when dealing with problems that have symmetry around an axis, making them ideal for surfaces that revolve around a central line, like a cylinder or a cone. This coordinate system uses three variables: \( r \), \( \theta \), and \( z \).

• \( r \) represents the radial distance from the central axis. It's analogous to the radius in polar coordinates and determines the distance of a point from the \( z \)-axis.
• \( \theta \) is the angular coordinate and is measured in the plane perpendicular to the \( z \)-axis. It shows the rotation angle around the axis and varies between \( 0 \) and \( 2\pi \).
• \( z \) denotes the height or the vertical distance from a reference plane, much like the \( z \)-coordinate in Cartesian coordinates.

Using cylindrical coordinates simplifies the equations for objects like cylinders and donut shapes (tori), since these objects naturally align with the symmetry of the cylindrical coordinate system. By expressing a surface through \( r = f(z) \), the equation becomes independent of \( \theta \), simplifying the analysis of symmetry.

Rotational Symmetry

Rotational symmetry occurs when a shape or object retains its overall appearance even after being rotated about an axis. This is a common feature in many natural and man-made structures, and it is a significant simplification in geometric problem-solving.In the case of the equation \( r = f(z) \) in cylindrical coordinates, the dependency of \( r \) only on \( z \) means for every specific height \( z \), \( r \) remains constant, making the distance from the \( z \)-axis uniform around a circle.

• This creates a rotationally symmetric surface around the \( z \)-axis.
• No matter how much the surface is rotated around this axis, it will look the same.

Understanding rotational symmetry is essential when analyzing surfaces in cylindrical coordinates because it often leads to simplifications in mathematical and physical descriptions.For example, equations of motion or light propagation in optics become easier to solve when the underlying surfaces or lenses possess rotational symmetry.

Geometric Transformations

Geometric transformations involve changing the position, size, or orientation of an object while retaining its shape and proportions. These transformations are fundamental in understanding different perspectives in geometry and algebra.In cylindrical coordinate systems, such transformations often include rotations and translations, which help in visualizing symmetry and simplifying equations.

• Rotations: Since the object in question has rotational symmetry, any rotation around the \( z \)-axis does not change the shape of the object, thereby preserving its geometry.
• Translations: Moving the entire surface up and down along the \( z \)-axis will not affect its symmetry or shape, as it only involves changing the \( z \) coordinate.
• Scaling: Uniformly increasing or decreasing \( r \) values leads to a proportional change in the object's size without altering its symmetry.

These transformations are not just mathematical curiosities but are essential tools in fields like computer graphics, robotics, and physics where understanding the position and orientation of objects is crucial.