Problem 14 - Calculus: Early Transcendentals

Question

We can describe points in the first octant by \(x>0, y>0\) and \(z>0 .\) Give similar inequalities for the first octant in cylindrical and spherical coordinates.

Step-by-Step Solution

Answer

Cylindrical: \(r > 0, 0 < \theta < \frac{\pi}{2}, z > 0\); Spherical: \(\rho > 0, 0 < \theta, \phi < \frac{\pi}{2}\).

Step 1: Converting to Cylindrical Coordinates

In cylindrical coordinates, points are described using coordinates \((r, \theta, z)\), where \(r\) is the radial distance from the \(z\)-axis, \(\theta\) is the angle around the \(z\)-axis, and \(z\) is the height along the \(z\)-axis. To be in the first octant, the conditions are: \(r > 0\), \(0 < \theta < \frac{\pi}{2}\), and \(z > 0\).

Step 2: Converting to Spherical Coordinates

In spherical coordinates, points are described using coordinates \((\rho, \theta, \phi)\), where \(\rho\) is the distance from the origin, \(\theta\) is the same azimuthal angle as in cylindrical coordinates, and \(\phi\) is the polar angle from the positive \(z\)-axis. For the first octant, the conditions are: \(\rho > 0\), \(0 < \theta < \frac{\pi}{2}\), and \(0 < \phi < \frac{\pi}{2}\).

Key Concepts

Cylindrical Coordinates

Cylindrical coordinates provide a practical way of describing a point in three-dimensional space. Instead of using just the Cartesian coordinates \(x, y, z\), cylindrical coordinates define a location using three variables: \( (r, \theta, z) \). Here's how each component works:

• \(r\): the radial distance from the z-axis, similar to the radius in a circle.
• \(\theta\): the angle formed with respect to the positive x-axis, measured counterclockwise.
• \(z\): the height or the same z-axis component as in Cartesian coordinates.

When a point is positioned in the first octant, it means it is in the positive regions of all three axes. Therefore, we need to adjust our inequalities:

• \(r > 0\): ensures the point is 'away' from the origin.
• \(0 < \theta < \frac{\pi}{2}\): restricts the angle so the point is in the first quadrant of the xy-plane.
• \(z > 0\): ensures the point is above the xy-plane.

Spherical Coordinates

Spherical coordinates offer another method to locate a point, emphasizing distances and angles. In spherical coordinates, a point is represented by \( (\rho, \theta, \phi) \). Each component has a specific role:

• \(\rho\): distance from the origin to the point, akin to radius in polar coordinates but in 3D.
• \(\theta\): the azimuthal angle, similar to the angle around the z-axis in cylindrical coordinates.
• \(\phi\): the polar angle, which is the angle from the positive z-axis down to the point.

To describe a point in the first octant using spherical coordinates, we use the following conditions:

• \(\rho > 0\): asserts that the point isn't at the origin.
• \(0 < \theta < \frac{\pi}{2}\): places the point within the first quadrant of the xy-plane.
• \(0 < \phi < \frac{\pi}{2}\): ensures the point is above the xy-plane, keeping the angle less than a right angle from the z-axis.

First Octant

The concept of the first octant arises from the division of 3D space into eight sections by the coordinate planes (xy-plane, yz-plane, and xz-plane). Each octant represents a region where the signs of the x, y, and z coordinates differ. The first octant is where all coordinates are positive.
In both cylindrical and spherical coordinates, the first octant corresponds to specific conditions. These ensure that all coordinates produced are in the positive sections. Remember:

• In cylindrical coordinates: \(r > 0, \ 0 < \theta < \frac{\pi}{2}, \ z > 0\).
• In spherical coordinates: \(\rho > 0, \ 0 < \theta < \frac{\pi}{2}, \ 0 < \phi < \frac{\pi}{2}\).

Each setup ensures that the spatial description remains in the intended positive space.

Mathematical Inequalities

Mathematical inequalities are expressions that depict the relative size or order of two objects. In coordinate systems, inequalities help define bounds or constraints, crucial for specifying positions geometrically.
In diagrams or analyses using cylindrical or spherical coordinates, inequalities guide us in honing in on specific regions, just like in this exercise for identifying points in the first octant.
Consider the key inequalities from the exercise:

• For both cylindrical and spherical representations, \(r > 0\) and \(\rho > 0\) keep the point distant from the origin.
• Angles \(\theta\) and \(\phi\) are restricted to less than \(< \frac{\pi}{2}\) to limit the points to positive x and y values.
• Finally, \(z > 0\) keeps points above the xy-plane, ensuring all coordinates remain in positive domains.