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) \(\rho=2, \quad 0 \leq \theta \leq 2 \pi, \quad \varphi=\pi / 6\)

Short answer

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

Step-by-step solution

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.

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.

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 \(\pi/4\).

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 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.