Problem 4 - Calculus Early Transcendentals

Question

Describe the set \(\\{(\rho, \varphi, \theta): \varphi=\pi / 4\\}\) in spherical coordinates.

Step-by-Step Solution

Answer

Answer: A right circular cone with a semi-vertical angle of 45 degrees centered around the z-axis.

Step 1: Visualize the problem

First, we need to understand the meaning of each coordinate in a spherical coordinate system. The spherical coordinate system (ρ, φ, θ) consists of: - ρ (rho): The distance from the origin to the point - φ (phi): The angle between the positive z-axis and the line connecting the origin to the point (also known as polar angle, with the range 0 ≤ φ ≤ π) - θ (theta): The angle between the x-axis and the projection of the point into the xy-plane (0 ≤ θ ≤ 2π)

Step 2: Interpret the given expression

We are given \(\varphi = \frac{\pi}{4}\), which means the angle between the positive z-axis and the line connecting the origin to the point is constant at \(\frac{\pi}{4}\) or 45 degrees.

Step 3: Visualize the geometry of the given set

Now, let's describe the geometry represented by this set. Since the polar angle is constant at 45 degrees, this means that all points of this set lie on a cone centered around the z-axis. The origin is the vertex of the cone and its semi-vertical angle is 45 degrees.

Step 4: Express the geometric description of the set

The set \(\{(\rho, \varphi, \theta): \varphi = \frac{\pi}{4}\}\) in spherical coordinates represents a right circular cone with a semi-vertical angle of 45 degrees centered around the z-axis.