Problem 17 - Thomas Calculus

Question

In Exercises \(13-22,\) sketch the graph described by the following spherical coordinates in three-dimensional space. $$ \rho \cos \phi=4 $$

Step-by-Step Solution

Answer

The graph is a horizontal plane at \( z = 4 \).

Step 1: Convert Spherical to Cartesian Coordinates

In spherical coordinates, a point is represented by \( (\rho, \theta, \phi) \), where \( \rho \) is the radial distance, \( \theta \) is the azimuthal angle, and \( \phi \) is the polar angle. The Cartesian coordinates \((x, y, z)\) are related to spherical coordinates by the following formulas: \[ x = \rho \sin \phi \cos \theta, \quad y = \rho \sin \phi \sin \theta, \quad z = \rho \cos \phi \]. We have the equation \( \rho \cos \phi = 4 \), which simplifies to \( z = 4 \).

Step 2: Analyze the Equation in Cartesian Coordinates

The equation \( z = 4 \) implies that every point on the surface has the \( z \)-coordinate equal to 4. Since there are no restrictions on \( \rho \) or \( \theta \), \((x, y)\) can be any real number, resulting in a horizontal plane at \( z = 4 \).

Step 3: Sketch the Graph

To sketch the graph, draw a horizontal plane parallel to the \( xy \)-plane in three-dimensional space, intersecting the \( z \)-axis at \( z = 4 \). The plane extends infinitely along the \( x \) and \( y \) directions.

Key Concepts

Cartesian Coordinates

In the world of geometry, Cartesian coordinates create a system that allows us to specify the location of points uniquely in a plane or space. This system is usually structured using axes that are at right angles to each other. In two dimensions, we have the familiar X and Y axes.
But in the three-dimensional space, an additional Z axis comes into play. When talking about converting from spherical to Cartesian coordinates, it's essential to understand that each point in spherical coordinates

• has a radial distance \( \rho \),
• an azimuthal angle \( \theta \),
• and a polar angle \( \phi \).

These correspond to specific equations in Cartesian coordinates:

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

For instance, in our example, having the equation \( \rho \cos \phi = 4 \) transforms directly into \( z = 4 \). This conversion process helps visualize how the point sits in the three-dimensional space, maintaining its location consistently.

Three-Dimensional Space

Three-dimensional space is the environment where our physical world exists. It represents all directions an object can move: length, width, and height. In mathematics, it is defined using three axes:

• The X-axis (horizontal)
• The Y-axis (another horizontal)
• The Z-axis (vertical)

Imagine a room. The floor stretches along the X and Y, while the walls or height represent the Z-axis.
This three-dimensional view is pivotal when dealing with objects that aren't flat and need an additional dimension for depth.To work with coordinates, the three-dimensional space uses the triplet

• (x, y, z),

where all three values together pinpoint the exact location. As we saw in the equation \( z = 4 \), the focus was on a plane in this space. The plane, like you might find as a level floor extending infinitely, shows how these coordinate systems help us understand the position within three-dimensional space.

Horizontal Plane

The aspect of a horizontal plane in three-dimensional space is what makes certain positions "flat". In our example, the plane is defined by the constant \( z = 4 \) in the Cartesian coordinate system. Imagine a surface like a perfectly flat tabletop or a calm sea horizon that extends without end in the X and Y axes.A horizontal plane maintains the same height across its surface. The definition \( z = 4 \) tells us the plane remains parallel to the XY-plane but is shifted upwards, indicating elevation in the three-dimensional space.

• The horizontal plane does not just confine to specific limits, unlike a square or rectangle.
• It continues infinitely as long as its height is fixed.

When sketching it, one represents it by drawing it parallel to the traditional grid but elevated at the point it intersects the Z-axis, as happens at our point \( z = 4 \). This horizontal plane serves various functions in engineering, physics, and even architecture by defining surfaces or boundaries where those elements are essential.