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Chapter 13: Problem 4

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

Short Answer

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

Step by step solution

01

Visualize the problem

First, we need to understand the meaning of each coordinate in a spherical coordinate system. The spherical coordinate system (蟻, 蠁, 胃) consists of: - 蟻 (rho): The distance from the origin to the point - 蠁 (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 鈮� 蠁 鈮� 蟺) - 胃 (theta): The angle between the x-axis and the projection of the point into the xy-plane (0 鈮� 胃 鈮� 2蟺)
02

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.
03

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.
04

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.

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Most popular questions from this chapter

A disk of radius \(r\) is removed from a larger disk of radius \(R\) to form an earring (see figure). Assume the earring is a thin plate of uniform density. a. Find the center of mass of the earring in terms of \(r\) and \(R\) (Hint: Place the origin of a coordinate system either at the center of the large disk or at \(Q\); either way, the earring is symmetric about the \(x\) -axis.) b. Show that the ratio \(R / r\) such that the center of mass lies at the point \(P\) (on the edge of the inner disk) is the golden mean \((1+\sqrt{5}) / 2 \approx 1.618\)

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