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Chapter 14: Problem 4

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

Short Answer

Expert verified
Question: Describe the set in spherical coordinates given the condition 蠁 = 蟺/4. Answer: The set can be described in spherical coordinates as: { (蟻, 蠁, 胃): 蟻 鈮� 0, 蠁 = 蟺/4, 胃 鈭� [0, 2蟺] }.

Step by step solution

01

Recall the definition of spherical coordinates

In spherical coordinates, a point \((\rho, \varphi, \theta)\) represents a point in space with radius \(\rho\), polar angle \(\varphi\), and azimuthal angle \(\theta\). Polar angle, \(\varphi\), ranges from \(0\) to \(\pi\), and azimuthal angle, \(\theta\), ranges from \(0\) to \(2\pi\).
02

Visualize the given condition

We are given the condition that \(\varphi = \pi / 4\). This implies that all the points in the set lie on a plane that makes an angle of \(\pi/4\) with the positive z-axis.
03

Describe the set in spherical coordinates

Due to the given condition, \(\varphi = \pi / 4\), every point in our set will have a polar angle of \(\pi/4\). The radius, \(\rho\), can be any non-negative value, and the azimuthal angle, \(\theta\), can range from \(0\) to \(2\pi\). Therefore, the set can be described in spherical coordinates as: \(\\{(\rho, \varphi, \theta): \rho \geq 0,\, \varphi = \pi / 4,\, \theta \in [0, 2\pi]\\}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radius
In the context of spherical coordinates, the radius, denoted by \( \rho \), is a measure of the distance from the origin to the point in space. Imagine it like a thread strung from the center of a sphere to any point on its surface. This distance can theoretically stretch to infinity, but for practical purposes, we often consider it within a certain space range.
In spherical coordinates:
  • \( \rho \) can be any non-negative value.
  • It directly influences how far the point is from the center along the radial vector.
Remember, in the sphere's context, even if two points have the same polar angle \( \varphi \) and azimuthal angle \( \theta \), differing values of \( \rho \) change their position radically. For example, a point with \( \rho = 3 \) lies farther from the origin than one with \( \rho = 1 \), given identical angles in spherical coordinates.
Polar Angle
The polar angle, represented as \( \varphi \), describes the angle between the position vector of a point and the positive z-axis. If you picture a globe, the polar angle is akin to the latitude but measured as the angle itself, in radians, and not in degrees.
Here's what you need to know:
  • \( \varphi \) ranges between \( 0 \) and \( \pi \) radians.
  • A polar angle of \( 0 \) radians points directly upwards along the z-axis, while \( \pi \) radians points directly downwards.
In the given exercise, a fixed polar angle of \( \varphi = \pi/4 \) signifies points lying on a conical surface which consistently maintains this angle with the z-axis, creating a symmetrical arrangement around the z-axis.
Azimuthal Angle
The azimuthal angle, denoted by \( \theta \), is crucial for determining a point's rotational position around the z-axis. It's best visualized as the direction around the sphere at any given polar angle, much like longitude lines circling a globe.
Several important aspects of \( \theta \) include:
  • \( \theta \) ranges from \( 0 \) to \( 2\pi \) radians, covering full rotation around the z-axis.
  • An azimuthal angle of \( 0 \) radians starts at the positive x-axis, moving counter-clockwise as it increases.
Ultimately, \( \theta \) determines where the point lies in the xy-plane for any specified radius and polar angle. So, in the exercise, regardless of \( \theta \)'s variations, fixing the polar angle ensures the points remain confined to a specific conical form while still freely rotating around the vertical axis.

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Most popular questions from this chapter

A thin plate is bounded by the graphs of \(y=e^{-x}, y=-e^{-x}, x=0,\) and \(x=L .\) Find its center of mass. How does the center of mass change as \(L \rightarrow \infty ?\)

A thin rod of length \(L\) has a linear density given by \(\rho(x)=\frac{10}{1+x^{2}}\) on the interval \(0 \leq x \leq L\) Find the mass and center of mass of the rod. How does the center of mass change as \(L \rightarrow \infty ?\)

Choose the best coordinate system and find the volume of the following solid regions. Surfaces are specified using the coordinates that give the simplest description, but the simplest integration may be with respect to different variables. The wedge cut from the cardioid cylinder \(r=1+\cos \theta\) by the planes \(z=2-x\) and \(z=x-2\)

A point mass \(m\) is a distance \(d\) from the center of a thin spherical shell of mass \(M\) and radius \(R\). The magnitude of the gravitational force on the point mass is given by the integral \(F(d)=\frac{G M m}{4 \pi} \int_{0}^{2 \pi} \int_{0}^{\pi} \frac{(d-R \cos \varphi) \sin \varphi}{\left(R^{2}+d^{2}-2 R d \cos \varphi\right)^{3 / 2}} d \varphi d \theta\), where \(G\) is the gravitational constant. a. Use the change of variable \(x=\cos \varphi\) to evaluate the integral and show that if \(d>R,\) then \(F(d)=\frac{G M m}{d^{2}},\) which means the force is the same as if the mass of the shell were concentrated at its center. b. Show that if \(d

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