Problem 18 - Thomas Calculus

Question

In Exercises \(13-22,\) sketch the graph described by the following spherical coordinates in three-dimensional space. $$ 1 \leq \rho \leq 2 \sec \phi, \quad 0 \leq \phi \leq \frac{\pi}{4} $$

Step-by-Step Solution

Answer

Visualize the region as a spherical wedge between a cone and expanding surface.

Step 1: Understanding Spherical Coordinates

Spherical coordinates \(\rho, \phi, \theta\) are used to describe points in three-dimensional space. Here, \(\rho\) is the radial distance from the origin, \(\phi\) is the angle measured from the positive z-axis, and \(\theta\) is the azimuthal angle in the xy-plane from the positive x-axis. For this problem, note that \(\theta\) is not restricted.

Step 2: Interpreting the Given Conditions

The given conditions \(1 \leq \rho \leq 2 \sec \phi\) and \(0 \leq \phi \leq \frac{\pi}{4}\) describe a region in spherical coordinates. The condition \(\rho \leq 2 \sec \phi\) describes a surface, and inside it, \(\rho\) must be at least 1.

Step 3: Equating Surfaces in Cartesian Form

Convert \(\rho \leq 2 \sec \phi\) to Cartesian coordinates for visualization. Recall \(\sec \phi = \frac{1}{\cos \phi}\). Hence, the boundary becomes \(\rho \cos \phi \leq 2\), which can be reinterpreted in Cartesian coordinates as \(z = \rho \cos \phi\) with \(\rho = \sqrt{x^2 + y^2 + z^2}\).

Step 4: Visualizing the Shape

Considering the bounds \(0 \leq \phi \leq \frac{\pi}{4}\) and \(1 \leq \rho \leq 2 \sec \phi\), this represents a region of a spherical wedge or segment which increases in radius as \(\phi\) decreases from \(\frac{\pi}{4}\) to 0. The region is widest when \(\phi = 0\) where \(\rho = 2\), tapering off as \(\phi\) increases.

Step 5: Sketching the Graph

Now, draw the sketch. Begin by sketching a cone formed when \(\phi = \frac{\pi}{4}\). The outer boundary \(\rho = 2 \sec \phi\) is slightly complex, but visualize and sketch it as expanding from \(\phi = 0\) to \(\phi = \frac{\pi}{4}\), creating a region within the defined conic exemplified by the given \(\phi\) and \(\rho\) limits.

Key Concepts

3D Graph Sketching

When learning about spherical coordinates, graph sketching is an essential skill to visualize and understand three-dimensional space. In spherical coordinates, \( \rho \)is the radial distance from the origin, \( \phi \) is the polar angle measured from the positive z-axis, and \( \theta \) is the azimuthal angle in the xy-plane from the positive x-axis.
This exercise involves interpreting spherical conditions, primarily focusing on \( \rho \leq 2 \sec \phi \) and \( 0 \leq \phi \leq \frac{\pi}{4} \). These inequalities define a specific region in three-dimensional space.

• Focus first on understanding boundary surfaces, represented mathematically.
• Cones and spherical measures are often involved, indicating radially symmetric shapes about the z-axis.

Envisioning these conditions requires translating abstract mathematical constraints into concrete shapes, such as cones or spheres. This tangible analogy facilitates the graph sketching process and aids the comprehension of spatial relations and boundaries.

Spherical to Cartesian Conversion

Converting spherical coordinates to Cartesian coordinates allows for easier visualization and manipulation of surfaces. The conversion formulas, which are:

• \( x = \rho \sin \phi \cos \theta \)
• \( y = \rho \sin \phi \sin \theta \)
• \( z = \rho \cos \phi \)

These transformations map spherical coordinates into the familiar Cartesian plane.
This conversion is essential when visualizing the inequality \( \rho \leq 2 \sec \phi \). Since \( \sec \phi = \frac{1}{\cos \phi} \), the conversion leads to the equation \( z \leq 2 \), provided by \( \rho \cos \phi = z \). This step takes abstract spherical parameters into a Cartesian context where geometric shapes become easier to comprehend and sketch.

Bounded Regions in Spherical Coordinates

In spherical coordinates, bounded regions signify areas confined between certain surfaces like cones and spheres.
Understanding the inequality \( 1 \leq \rho \leq 2 \sec \phi \), regional boundaries become apparent:

• The lower boundary is a spherical shell where any point has a minimum distance of 1 unit from the origin.
• The upper boundary, dictated by \( \rho \leq 2 \sec \phi \), creates a region extending outward under the cap of \( 2 \sec \phi \).

The limits on \( \phi \), from 0 to \( \frac{\pi}{4} \), further contain the region within a specific angular wedge of space.
This setup hints at a sector of space limited not just in radial distance but also in angular sweep, visualizing a bendable and adjustable slice of space, effectively a mechanical model of spherical constraints.

Visualizing Spherical Surfaces

Visualizing spherical surfaces involves understanding the influences of \( \rho, \phi, \)and \( \theta \) on creating these forms in space. The conditions \( 1 \leq \rho \leq 2 \sec \phi \) and \( 0 \leq \phi \leq \frac{\pi}{4} \) are vital to defining the shape and positioning in a three-dimensional realm.

• \( \phi \) determines the angle within the z-axis, dictating whether the region opens wide or narrows.
• The limit of \( \phi \) ensures the region remains within a finite sector, akin to a pie slice.
• The condition on \( \rho \) manages the distance from the origin, forming the boundary and depth of the sector.

Visualizing these surfaces effectively means recognizing how the mathematical parameters dictate spatial form. The boundaries are flexible yet restricted, forming a zoned-in spherical slice whose boundaries shift with changes in \( \phi \) and \( \rho \).