Problem 14 - Thomas Calculus

Question

In Exercises \(13-22,\) sketch the graph described by the following spherical coordinates in three-dimensional space. $$ \phi=\frac{\pi}{6} $$

Step-by-Step Solution

Answer

The graph is a cone with vertex at the origin, axis along the \( z \)-axis, and apex angle of \( 60^\circ \).

Step 1: Understanding Spherical Coordinates

In spherical coordinates, a point in 3D space is defined by the coordinates \( (r, \theta, \phi) \). Here, \( r \) is the radial distance from the origin, \( \theta \) is the azimuthal angle in the \( xy \)-plane from the positive \( x \)-axis, and \( \phi \) is the polar angle from the positive \( z \)-axis.

Step 2: Given Value of \( \phi \)

In this exercise, we are given \( \phi = \frac{\pi}{6} \). This indicates that the angle from the positive \( z \)-axis to the point is \( 30^\circ \), since \( \frac{\pi}{6} \) radians is equivalent to \( 30^\circ \).

Step 3: Visualizing Lines of Constant \( \phi \)

When \( \phi \) is constant, it describes a cone with its vertex at the origin, apex angle \( 2\phi \) (or \( 60^\circ \) here), aligned along the \( z \)-axis. All points satisfying this have the same angle \( 30^\circ \) from the \( z \)-axis.

Step 4: Sketching the Graph

Draw the \( xyz \)-coordinate system. From the origin, sketch a cone with its axis coinciding with the \( z \)-axis. The cone has its surface at an angle \( \frac{\pi}{6} \) from the \( z \)-axis, so its apex angle is \( 60^\circ \).

Key Concepts

Three-dimensional space

In the realm of mathematics and physics, three-dimensional space refers to a space in which three values are required to determine the position of an element. This space is what we commonly experience in our daily lives. When we talk about three-dimensional space, we're referring to a world where objects have depth, width, and height.

The coordinates in this type of space are usually represented as \((x, y, z)\) in Cartesian coordinates. However, we can also describe locations using spherical coordinates, where points are given by \((r, \theta, \phi)\).

• \(r\): Radial distance from the origin to the point.
• \(\theta\): Azimuthal angle, usually measured in the \(xy\)-plane from the positive \(x\)-axis.
• \(\phi\): Polar angle from the positive \(z\)-axis.

This system is especially useful for describing positions relative to a central point, such as with planets orbiting a star.

Polar angle

The polar angle \(\phi\) in spherical coordinates measures the angle between the positive \(z\)-axis and the line to the point. It ranges from \(0\) to \(\pi\) radians, covering angles from directly upwards from the \(z\)-axis to directly downwards.

In this particular problem, we have \(\phi = \frac{\pi}{6}\), which translates to an angle of \(30^{\circ}\). This means that any point on this sphere lies at a \(30^{\circ}\) angle from the \(z\)-axis.

Visualizing this, one can imagine setting a protractor at the origin and measuring \(30^{\circ}\) down from the \(z\)-axis. All points with this polar angle will form a conical shape in three-dimensional space, with the cone's surface at this consistent angle from the \(z\)-axis. The full apex angle of the cone would be \(60^{\circ}\), as described in the solution above.

Azimuthal angle

The azimuthal angle \(\theta\), in the system of spherical coordinates, is the angle measured in the \(xy\)-plane from the positive \(x\)-axis. It can vary from 0 to \(2\pi\) radians, encompassing full rotation around the \(z\)-axis.

While the azimuthal angle \(\theta\) wasn't specifically given in this exercise, it's crucial to understanding how a point or set of points can be distributed around the \(z\)-axis. When drawing the shapes described by spherical coordinates, \(\theta\) helps in determining in which direction on the horizontal plane the point lies.

• A point with \(\theta = 0\) lies directly on the positive \(x\)-axis.
• As \(\theta\) increases, the point revolves around the \(z\)-axis.

Understanding \(\theta\) is essential for graphing points in three-dimensional space, especially in applications such as physics and engineering.

Graph sketching

Graph sketching in spherical coordinates involves translating the mathematical description of a point or shape into a visual illustration. The goal is to bring clarity to the relationships between the different coordinate values.

In the case of our exercise, when \(\phi = \frac{\pi}{6}\), we are tasked with drawing a cone in three-dimensional space. The vertex of the cone starts at the origin, extending outward with a surface formed at a \(30^{\circ}\) angle from the \(z\)-axis.

• Start by positioning the \(xyz\) coordinate system clearly on paper.
• Then, draw a straight line representing the \(z\)-axis.
• From the origin, delineate a circular path at \(\phi = \frac{\pi}{6}\), directly translating this into the slant of the cone.

Understandably, in graph sketching, envisioning the spatial layout is key to accurate and meaningful representations. Tying angles into spatial orientations enhances problem-solving skills and conceptual understanding.