Problem 14 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of the given cylindrical or spherical equation. $$ \rho=\sec \phi $$

Step-by-Step Solution

Answer

The graph is a cylindrical surface symmetric about the z-axis.

Step 1: Understand the Spherical Coordinate System

The given equation, \( \rho = \sec \phi \), is in spherical coordinates. In this coordinate system, \( \rho \) is the radial distance from the origin, \( \phi \) is the polar angle (angle with the positive z-axis), and \( \theta \) is the azimuthal angle. The range for \( \phi \) is from 0 to \( \pi \).

Step 2: Recall Relationship of Secant Function

The secant function is the reciprocal of the cosine function, so \( \sec \phi = \frac{1}{\cos \phi} \). Thus, \( \rho = \sec \phi = \frac{1}{\cos \phi} \). This implies that the equation becomes \( \rho \cos \phi = 1 \).

Step 3: Analyze the Implied Condition

From the previous step, the equation \( \rho \cos \phi = 1 \) implies a relationship between \( \rho \) and \( \phi \). In a context where \( \rho = 1/\cos \phi \), \( \phi \) must be strictly in the domain where \( \cos \phi eq 0 \), which excludes \( \phi = \frac{\pi}{2} \).

Step 4: Determine the Shape of the Surface

The equation \( \rho \cos \phi = 1 \) indicates that for each angle \( \phi \), \( \rho \) grows as \( \phi \) approaches \( \frac{\pi}{2} \) from either side. This builds a surface where for every \( \theta \), \( \rho \) is determined entirely by \( \phi \), describing a cylinder in the polar angle direction.

Step 5: Sketch the Graph

To sketch the graph, note that the surface extends along all azimuthal angles \( \theta \) and consists of the collection of points where \( \rho \cos \phi = 1 \). This is a cylindrical surface symmetric about the z-axis, excluding the section around \( \phi = \frac{\pi}{2} \).

Key Concepts

Radial Distance

In spherical coordinates, the radial distance, denoted as \( \rho \), represents the distance from the origin to a point in space. This is similar to the radius in a spherical object, reaching from the center to its surface. Radial distance is always non-negative.
In the given exercise, \( \rho = \sec \phi \), meaning the radial distance is dependent on the polar angle \( \phi \). This relationship might seem complex, but remember that \( \rho \) helps define the geometry of the surface we are studying.
Understanding \( \rho \) is crucial, as it maps a point's position relative to the origin in spherical settings.

Polar Angle

The polar angle, represented by \( \phi \), is the angle measured from the positive z-axis downward towards the xy-plane. It ranges from 0 to \( \pi \) radians. This angle helps pinpoint locations in the spherical coordinate system.
In the equation \( \rho = \sec \phi \), \( \phi \) dictates how far the radial distance extends from the origin. As \( \phi \) changes, it affects \( \rho \) through the secant function, causing variations in the point's radial distance.
We must keep \( \phi \) from being \( \frac{\pi}{2} \) since \( \cos \phi = 0 \) there, which causes the secant to become undefined.

Azimuthal Angle

The azimuthal angle is denoted by \( \theta \). This angle measures the rotation around the z-axis and extends from 0 to \( 2\pi \). Think of it as the angle similar to the hands of a clock turning around the dial.
Though \( \theta \) isn't directly part of the equation \( \rho = \sec \phi \), it is implied as the surface extends uniformly across all azimuthal angles. Understanding \( \theta \) helps visualize how this cylindrically shaped surface can be envisioned around the 3D space, rotating around the origin.
Each value of \( \theta \) corresponds to a full 360° sweep around the axis, contributing to the symmetry of the structure.

Secant Function

The secant function is a trigonometric function related to the cosine function. Specifically, it is defined as \( \sec \phi = \frac{1}{\cos \phi} \). This relationship reveals how the secant function behaves, especially around certain critical points.
In our context, \( \rho = \sec \phi \) implies that when \( \cos \phi \) approaches zero, \( \sec \phi \) becomes very large. Therefore, care must be taken to avoid angles where \( \cos \phi = 0 \), like \( \phi = \frac{\pi}{2} \).
This secant feature drastically affects \( \rho \), leading to fascinating surface structures where distance scales with \( \phi \). The secant function's properties allow for the emergence of cylindrical surfaces in 3D space.

Cylindrical Surface

The concept of a cylindrical surface comes alive in this exercise. In a 3D spherical setting, it's not just about circular bases but extends straight along the height or axis.
In the problem, the cylindrical surface is formed by the condition \( \rho = \sec \phi \). This condition results in infinite extensions along the azimuthal angle \( \theta \), plotting out a cylinder around the z-axis.
The symmetry here lies in consistent radial distance changes with \( \phi \), covering a complete rotational perimeter. Note that this surface excludes \( \phi = \frac{\pi}{2} \), as \( \sec \phi \) would be undefined.