Problem 16 - Thomas Calculus

Question

In Exercises \(13-22,\) sketch the graph described by the following spherical coordinates in three-dimensional space. $$ \rho=\csc \phi $$

Step-by-Step Solution

Answer

The graph is a horn-shaped surface created by circles of varying radii as \( \phi \) changes.

Step 1: Identify the Given Spherical Equation

We are given the spherical equation \( \rho = \csc \phi \). In spherical coordinates, \( \rho \) is the radial distance, \( \theta \) is the azimuthal angle, and \( \phi \) is the polar angle. The equation relates \( \rho \) to \( \phi \).

Step 2: Express \( \rho \) Using \( \phi \)

The equation \( \rho = \csc \phi \) implies that \( \rho = \frac{1}{\sin \phi} \). This means \( \rho \) is the reciprocal of the sine of the polar angle, \( \phi \).

Step 3: Understand \( \csc \phi \) Behavior

The function \( \csc \phi = \frac{1}{\sin \phi} \) will approach infinity as \( \phi \) approaches multiples of \( \pi \), where \( \sin \phi = 0 \). However, in spherical coordinates, \( \phi \) typically ranges from 0 to \( \pi \).

Step 4: Determine General Shape and Behavior

Since \( \rho \) changes with \( \phi \) and is undefined when \( \sin \phi = 0 \), the surface described by this equation will have a different radius of \( \frac{1}{\sin \phi} \) on different 'slices' of constant \( \theta \).

Step 5: Sketching the Surface

With \( \rho \) defined in terms of \( \phi \), and \( \theta \) being free since it does not appear in the equation, this surface is a collection of circles stacked at different radii. As \( \phi \) varies from 0 to \( \pi \), the radius \( \rho \) varies, creating a horn-shaped surface.

Key Concepts

Graph Sketching

Graph sketching in the context of spherical coordinates involves translating mathematical equations into visual representations in three-dimensional space. It is akin to sketching 2D graphs but with an added dimension for depth. When sketching the graph for a given spherical equation like \( \rho = \csc \phi \), we need to understand how each component of the spherical coordinates works together to form shapes in 3D.
Start by identifying the role of each variable. Here, \( \rho \) indicates the radial distance from the origin, and since \( \rho = \csc \phi \), it changes with the angle \( \phi \). \( \theta \) is not present in our equation, suggesting that it is free, serving as a rotational symmetry around the vertical axis. The sketch requires visualizing circles or rings changing in radius as they "stack" over changes in \( \phi \), resulting in unique geometric shapes.

Three-Dimensional Space

Three-dimensional (3D) space consists of three axes: typically labeled as x, y, and z. However, when dealing with spherical coordinates, we use \( \rho \), \( \theta \), and \( \phi \) to describe a point's location.
In 3D space, the position of an object is defined by distance from the origin (given by \( \rho \)), the azimuthal angle (\( \theta \), which measures the angle along the horizontal plane), and the polar angle (\( \phi \), measured from the positive z-axis downwards). These coordinates are essential for tackling problems involving volumes and surfaces because they offer a more natural way to describe objects like spheres or circular surfaces like those found in nature. Understanding how to interpret these in their x, y, z forms using the transformation formulas is a vital part of visualizing these problems.

Trigonometric Functions

Trigonometric functions play a crucial role in understanding equations in spherical coordinates. They allow us to relate angles to distances. In our case, \( \rho = \csc \phi \) implies \( \rho = \frac{1}{\sin \phi} \). This shows that \( \rho \) is dependent on the sine of angle \( \phi \).
Some important things to remember about trigonometric functions include:

• The sine function, \( \sin \phi \), ranges from -1 to 1, but in spherical coordinate contexts \( \phi \) is taken from 0 to \( \pi \), ensuring sine values are always positive.
• The cosecant function, \( \csc \phi \), is the reciprocal of sine, which becomes undefined exactly where sine is zero, highlighting particular problematic angles (multiples of \( \pi \)).
• These functions allow us to represent circular and spherical relationships effectively, used in many physical applications like wave behavior or electromagnetic fields.

Spherical Equations

Spherical equations like \( \rho = \csc \phi \) describe objects in 3D space using spherical coordinates rather than Cartesian. These equations provide insights into the form and structure of geometric surfaces.
In the case of \( \rho = \csc \phi \), the equation creates a surface that varies as \( \phi \) changes. The surface forms a series of circular layers at increments defined by \( \phi \). As \( \phi \) moves from 0 to \( \pi \), this alters \( \rho \) and thus the radius of each circle or ring in the figure.

• The beauty of spherical equations is in their ability to elegantly define complex structures using compact mathematical expressions.
• These relationships assist greatly in fields like astronomy and physics where modeling moving bodies and fields in 3D is critical.
• Understanding these can help us visualize phenomena ranging from planetary motion to molecular structures.