Problem 21 - Thomas Calculus

Question

In Exercises \(13-22,\) sketch the graph described by the following spherical coordinates in three-dimensional space. $$ \begin{array}{ll}{0 \leq \rho \cos \theta \sin \phi \leq 2,} & {0 \leq \rho \sin \theta \sin \phi \leq 3} \\ {0 \leq \rho \cos \phi \leq 4}\end{array} $$

Step-by-Step Solution

Answer

The inequalities describe a rectangular box in 3D space extending from the origin.

Step 1: Understanding Spherical Coordinates

Spherical coordinates \( (\rho, \theta, \phi) \) are another way of representing points in 3-dimensional space. Here, \( \rho \) represents the distance from the origin, \( \theta \) is the angle in the xy-plane from the positive x-axis, and \( \phi \) is the angle from the positive z-axis down towards the xy-plane.

Step 2: Translate Inequalities to Cartesian Form

Transform the given spherical inequalities to their Cartesian counterparts. The conversions from spherical to Cartesian are: \( x = \rho \cos \theta \sin \phi, \) \( y = \rho \sin \theta \sin \phi, \) and \( z = \rho \cos \phi. \) Apply these transformations to each inequality. We get: - \( 0 \leq x \leq 2 \)- \( 0 \leq y \leq 3 \)- \( 0 \leq z \leq 4 \)

Key Concepts

Three-dimensional space

Understanding three-dimensional space is crucial when dealing with spherical coordinates. This type of space includes all the points that exist along three axes: the x-axis, y-axis, and z-axis. In three-dimensional space, every position can be defined with three numbers that correspond to these axes.

• x-axis: Represents the horizontal dimension.
• y-axis: Denotes the vertical dimension.
• z-axis: Adds depth to the representation, creating a 3D effect.

Having a three-dimensional space allows you to represent real-world objects completely, such as a sphere or a cube, by covering every possible direction. This is quite different from two-dimensional space, which only covers length and width, lacking depth. In mathematics, especially in geometry and physics, being able to visualize or graphically represent data in three-dimensional space helps in understanding and solving complex problems. For instance, 3D models like graphs and vectors provide insights not visible in 2D space.

Cartesian coordinates

Cartesian coordinates are fundamental in mapping the position of points in three-dimensional space. They use three values, usually denoted as x, y, and z, to define a point's location based on its distance from each of three perpendicular axes. These axes intersect at a central point known as the origin.
The conversion from spherical to Cartesian coordinates translates a point in space from being defined by angles and a radial distance to the well-known grid system of straight lines. Here's how it works:

• x-coordinate: Obtained using the formula: \( x = \rho \cos \theta \sin \phi \).
• y-coordinate: Determined by: \( y = \rho \sin \theta \sin \phi \).
• z-coordinate: Given by: \( z = \rho \cos \phi \).

By employing these transformations, you can better understand how spherical coordinates define positions in 3D by transforming them into Cartesian coordinates. This is handy for graphing or sketching objects, providing a straightforward way to delineate the various boundaries and constraints given in equations.

Coordinate transformation

Coordinate transformation is the process of converting a point or a set of points from one coordinate system to another. In this exercise, performing a transformation from spherical coordinates to Cartesian coordinates helps in visualizing and sketching the described graph. Such transformations rely on mathematical formulas that adjust the way we interpret distances and angles.
Why transform coordinates? This method is essential because different types of coordinates offer varying advantages for specific tasks. Spherical coordinates make it easy to express equations that involve symmetry around a central point, like spheres. Cartesian coordinates, on the other hand, simplify linear equations and are preferred for graph plotting because they map directly onto standard grid paper.

• Advantages of Spherical Coordinates: Great for dealing with objects and movements around a sphere, such as celestial bodies.
• Benefits of Cartesian Coordinates: Ideal for detailed engineering or architectural designs that require precision.

Applying coordinate transformations ensures that you can leverage the strengths of both systems as needed, making complex mathematical and graphical tasks more approachable.