Problem 8

Question

Describe the curve, surface or region in space determined by the given bounds. Bounds in cylindrical coordinates: (a) $1 \leq r \leq 2, \quad \theta=\pi / 2, \quad 0 \leq z \leq 1$ (b) $r=2, \quad 0 \leq \theta \leq 2 \pi, \quad z=5$ Bounds in spherical coordinates: (c) $0 \leq \rho \leq 2, \quad 0 \leq \theta \leq \pi, \quad \varphi=\pi / 4$ (d) $\rho=2, \quad 0 \leq \theta \leq 2 \pi, \quad \varphi=\pi / 6$

Step-by-Step Solution

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Answer

(a) Annular slice of a cylinder; (b) Horizontal circle at $z=5$; (c) Cone sector; (d) Spherical circle section.

1Step 1: Interpret the Cylindrical Bounds for Part (a)

The given bounds in cylindrical coordinates are $1 \leq r \leq 2$, $\theta=\pi / 2$, and $0 \leq z \leq 1$. These describe a vertical strip or slice along the plane where $\theta = \pi/2$. The range $1 \leq r \leq 2$ indicates it is the annular region between two concentric circles, and $0 \leq z \leq 1$ restricts this region to lie between the planes $z=0$ and $z=1$. This forms an annular slice of a cylinder along the x-axis.

2Step 2: Interpret the Cylindrical Bounds for Part (b)

The bounds $r=2$, $0 \leq \theta \leq 2 \pi$, and $z=5$ describe a horizontal circular ring. The condition $r=2$ indicates a fixed radius, $0 \leq \theta \leq 2 \pi$ describes the complete circle, and $z=5$ places it in the plane $z = 5$. This forms a circle of radius 2 at the height $z=5$ in three-dimensional space.

3Step 3: Interpret the Spherical Bounds for Part (c)

The given bounds $0 \leq \rho \leq 2$, $0 \leq \theta \leq \pi$, and $\varphi=\pi / 4$ describe a cone section. Here, $\rho$ is the radial distance, $\theta$ the azimuthal angle, and $\varphi$ the polar angle. $\varphi=\pi/4$ fixes the angle of elevation from the polar axis, creating a planar disk section between heights $\rho \cos(\pi / 4)$. This gives a cone sector from the origin with a half-angle of $\pi/4$.

4Step 4: Interpret the Spherical Bounds for Part (d)

The bounds $\rho=2$, $0 \leq \theta \leq 2 \pi$, and $\varphi=\pi / 6$ describe a spherical circle. $\rho=2$ maintains a fixed radial distance, creating a spherical surface, $\varphi=\pi / 6$ specifies a fixed polar angle from the vertical, and $0 \leq \theta \leq 2 \pi$ completes the circle around the azimuth, determining a horizontal circular section on the sphere surface.

Key Concepts

Spherical CoordinatesRadiusPolar AngleAzimuthal Angle

Spherical Coordinates

Spherical coordinates offer an elegant way to describe points in a three-dimensional space using three distinct parameters: the radius (often denoted as $ \rho $), the polar angle ($ \varphi $), and the azimuthal angle ($ \theta $).

The unique feature of spherical coordinates is their suitability for systems and shapes that exhibit radial symmetry, such as spheres or globes.

$ \rho $ - Represents the radial distance from the origin to the point. It is always a non-negative value.
$ \varphi $ - The polar angle, measured from the positive z-axis, typically ranges from 0 to $ \pi $ radians.
$ \theta $ - The azimuthal angle, measured in the x-y plane from the positive x-axis. This angle usually ranges between 0 to $ 2\pi $ radians.

These coordinates are particularly useful in physics and engineering, where spherical symmetries are common. The transformation from spherical to Cartesian coordinates is established through the relationships:

$ x = \rho \sin \varphi \cos \theta $
$ y = \rho \sin \varphi \sin \theta $
$ z = \rho \cos \varphi $

This conversion helps in visualizing and solving problems that involve spherical regions or volumes.

Radius

In the context of spherical coordinates, the term "radius" refers to the distance $ \rho $ from the origin of the coordinate system to a given point in space. Think of it as the straight-line distance you'd measure with a ruler if you could stretch from the center out to the point of interest.

This radius is a crucial component as it effectively determines the position of a point in a three-dimensional space, signifying how far the point is from the center, or origin, of the sphere.

When $ \rho $ is 0, the point is at the origin.
As $ \rho $ increases, the point moves further away from the origin.

In applications like the bounds given in the original problems, $ \rho $ can simply restrict the boundary of a region, such as a cone or a spherical shell. A fixed $ \rho $ describes a surface at that specific distance from the origin, forming a sphere when combined with other angles.

Polar Angle

The polar angle, denoted as $ \varphi $ in spherical coordinates, plays an essential role in positioning a point within the three-dimensional coordinate system. This angle is measured from the positive z-axis, which often serves as a reference in such configurations.

Understanding this angle helps define the elevation of a point, which is especially useful in determining the shape and boundary of three-dimensional objects like cones and spheres.

A polar angle of 0 means the point is directly above the origin, lying along the positive z-axis.
A polar angle of $ \pi/2 $ indicates a point on the x-y plane.
The value of $ \pi $ places the point directly opposite to the initial starting point on the z-axis, below the x-y plane.

This concept allows for flexible and comprehensive mapping of points and regions, especially when paired with the radius and azimuthal angle, to form complete descriptions of three-dimensional spaces.

Azimuthal Angle

The azimuthal angle, represented by $ \theta $ in spherical coordinates, is integral for describing the orientation of a point in the x-y plane. This angle is measured from the positive x-axis toward the y-axis, wrapping around the origin. Think of it as the horizon direction or the "compass angle."

This concept of azimuthal angle adds the rotational aspect around a vertical axis, completing the full description of a point's location on spherical surfaces.

$ \theta = 0 $ situates the point on the positive x-axis.
$ \theta = \pi/2 $ aligns the point along the positive y-axis.
$ \theta = \pi/3 $ places it 60 degrees from the x-axis in the counter-clockwise direction, and so forth.

In determining the bounds of regions, this angle can delineate the extent of a region's rotation, thereby carving out specific segments within a spherical or circular framework. It's particularly beneficial when combined with other spherical coordinates to define complex geometries such as spherical caps or circular sections on spheres.

Problem 7

Problem 8

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